[{"data":1,"prerenderedAt":8493},["ShallowReactive",2],{"nav:algorithms":3,"lesson:\u002Falgorithms\u002Fintractability\u002Fnp-completeness":374,"course-wordcounts":8361,"ref-card-index":8417},[4,28,50,71,120,152,205,230,286,306,331,352],{"module":5,"moduleNumber":6,"slug":7,"lessons":8},"Foundations",1,"foundations",[9,15,21],{"title":10,"path":11,"lessonNumber":6,"topics":12,"summary":14},"What Is an Algorithm?","\u002Falgorithms\u002Ffoundations\u002Fwhat-is-an-algorithm",[5,13],"Correctness & Induction","An algorithm is a finite, mechanical recipe that transforms inputs into outputs. We pin down what counts as an algorithm, how we write one down, and the three things we always ask of it: is it correct, is it fast, and can we prove it.\n",{"title":16,"path":17,"lessonNumber":18,"topics":19,"summary":20},"Asymptotic Analysis","\u002Falgorithms\u002Ffoundations\u002Fasymptotic-analysis",2,[16],"We measure an algorithm's running time as a function of its input size, then strip away machine-specific constants and lower-order terms to compare algorithms cleanly. This lesson defines the RAM model and the $O$, $\\Omega$, $\\Theta$, $o$, and $\\omega$ notations, and shows how to read the cost of loops off the page.\n",{"title":22,"path":23,"lessonNumber":24,"topics":25,"summary":27},"Recurrences and the Master Theorem","\u002Falgorithms\u002Ffoundations\u002Frecurrences",3,[26],"Recurrences","Recursive and divide-and-conquer algorithms describe their own running time with a recurrence: $T(n)$ in terms of $T$ on smaller inputs. We solve recurrences three ways — drawing the recursion tree, guessing-and-verifying by induction, and applying the Master Theorem — using merge sort as the running example.\n",{"module":29,"moduleNumber":18,"slug":30,"lessons":31},"Divide & Conquer","divide-and-conquer",[32,38,44],{"title":33,"path":34,"lessonNumber":6,"topics":35,"summary":37},"Divide and Conquer & Mergesort","\u002Falgorithms\u002Fdivide-and-conquer\u002Fmergesort",[29,36],"Comparison Sorting","Divide and conquer breaks a problem into smaller copies of itself, solves them recursively, and stitches the answers together. We meet the paradigm through mergesort — its merge step, its loop-invariant proof, and the recursion tree that pins its cost at $\\Theta(n\\log n)$ — then glimpse Karatsuba multiplication as a second example of the same idea.",{"title":39,"path":40,"lessonNumber":18,"topics":41,"summary":43},"Quicksort","\u002Falgorithms\u002Fdivide-and-conquer\u002Fquicksort",[36,42],"Probabilistic Analysis","Quicksort sorts in place by partitioning around a pivot and recursing on each side. We give Lomuto and Hoare partitioning with a correctness invariant, see why a bad pivot costs $\\Theta(n^2)$ while a balanced one gives $\\Theta(n\\log n)$, and prove that randomizing the pivot makes the expected cost $\\Theta(n\\log n)$ on every input.",{"title":45,"path":46,"lessonNumber":24,"topics":47,"summary":49},"Linear-Time Selection","\u002Falgorithms\u002Fdivide-and-conquer\u002Fselection",[48,29],"Order Statistics","Finding the $k$-th smallest element looks like it should require sorting, but it does not. Quickselect adapts quicksort's partition to recurse on just one side, achieving expected $O(n)$. The median-of-medians algorithm guarantees a good pivot with the groups-of-five trick, pushing the worst case down to a provable $O(n)$.",{"module":51,"moduleNumber":24,"slug":52,"lessons":53},"Sorting & Order Statistics","sorting",[54,60,65],{"title":55,"path":56,"lessonNumber":6,"topics":57,"summary":59},"Heaps and Heapsort","\u002Falgorithms\u002Fsorting\u002Fheaps-and-heapsort",[58,36],"Heaps","A binary heap is a tree we store flat in an array, with index arithmetic standing in for pointers. We build the max-heap property bottom-up in $O(n)$ time, sort in place in $\\Theta(n\\log n)$ by repeatedly extracting the maximum, and reuse the same structure to implement a priority queue.",{"title":61,"path":62,"lessonNumber":18,"topics":63,"summary":64},"Lower Bounds for Comparison Sorting","\u002Falgorithms\u002Fsorting\u002Fsorting-lower-bounds",[36],"Every sort we have seen runs in $\\Omega(n\\log n)$, and that is no accident. Modeling a sort as a decision tree of comparisons, we show any such tree must have $n!$ leaves, forcing height $\\ge \\log_2(n!) = \\Omega(n\\log n)$ — a worst-case bound no comparison sort can ever beat.",{"title":66,"path":67,"lessonNumber":24,"topics":68,"summary":70},"Sorting in Linear Time","\u002Falgorithms\u002Fsorting\u002Flinear-time-sorting",[69],"Linear-Time Sorting","The $\\Omega(n\\log n)$ barrier only binds algorithms that compare. By instead using keys as array indices we slip past it: counting sort runs in $\\Theta(n+k)$ and is stable, radix sort layers it digit by digit, and bucket sort averages $\\Theta(n)$ on uniform data. We see exactly when each applies.",{"module":72,"moduleNumber":73,"slug":74,"lessons":75},"Data Structures",4,"data-structures",[76,82,88,93,99,105,113],{"title":77,"path":78,"lessonNumber":6,"topics":79,"summary":81},"Elementary Data Structures","\u002Falgorithms\u002Fdata-structures\u002Felementary-structures",[80],"Linear Structures","Every container is built one of two ways: **contiguous** in an array, or **linked** through pointers. We trade cache-friendly random access against $O(1)$ splicing, derive the **amortized $O(1)$** append of a doubling dynamic array, and assemble the two ordered access disciplines — the LIFO **stack** and the FIFO **queue** (with its generalization, the **deque**) — on top of both.",{"title":83,"path":84,"lessonNumber":18,"topics":85,"summary":87},"Hash Tables","\u002Falgorithms\u002Fdata-structures\u002Fhash-tables",[86],"Hashing","A hash table implements the dictionary — insert, search, delete — in expected $O(1)$ time by scattering keys across an array with a hash function. We build up from direct addressing, handle collisions by chaining and by open addressing, analyze the load factor $\\alpha$, and see how universal hashing earns its expected-time guarantee against every input.",{"title":89,"path":90,"lessonNumber":24,"topics":91,"summary":92},"Binary Search Trees","\u002Falgorithms\u002Fdata-structures\u002Fbinary-search-trees",[89],"A binary search tree keeps keys ordered so that every operation follows a single root-to-leaf path. We state the BST property, give search and insert, find minimum, maximum, and successor, see that an inorder walk emits the keys in sorted order, and confront the catch — every operation costs $O(h)$, and a carelessly built tree degrades to height $h = \\Theta(n)$, motivating balance.",{"title":94,"path":95,"lessonNumber":73,"topics":96,"summary":98},"AVL Trees","\u002Falgorithms\u002Fdata-structures\u002Favl-trees",[97],"Balanced Trees","An AVL tree is the first balanced BST: at every node the two subtrees' heights differ by at most $1$. A Fibonacci-style minimal-node argument forces height $h \\le 1.44\\log_2 n = O(\\log n)$, so search, insert, and delete are all $O(\\log n)$. Insertion rebalances with at most one of four rotation cases (LL, RR, LR, RL); deletion may rotate all the way to the root.",{"title":100,"path":101,"lessonNumber":102,"topics":103,"summary":104},"Balanced Search Trees","\u002Falgorithms\u002Fdata-structures\u002Fbalanced-trees",5,[97],"An ordinary BST can degrade to height $\\Theta(n)$; balanced search trees guarantee $h = O(\\log n)$ by maintaining invariants and repairing them after every update. We meet rotations, the local restructuring primitive, then red-black trees, whose color invariants force logarithmic height, and finally B-trees, which trade tall-and-thin for short-and-wide to win on disk.",{"title":106,"path":107,"lessonNumber":108,"topics":109,"summary":112},"Disjoint Sets (Union-Find)","\u002Falgorithms\u002Fdata-structures\u002Funion-find",6,[110,111],"Disjoint Sets","Amortized Analysis","The disjoint-set data structure tracks a partition of elements into groups, answering \"are these two in the same group?\" and merging groups on demand. A forest of parent pointers, sped up by union by rank and path compression, drives every operation to near-constant $O(\\alpha(n))$ amortized time — the engine behind connectivity queries and Kruskal's minimum spanning tree.",{"title":114,"path":115,"lessonNumber":116,"topics":117,"summary":119},"Fenwick & Segment Trees","\u002Falgorithms\u002Fdata-structures\u002Ffenwick-and-segment-trees",7,[118],"Range Queries","A prefix-sum array answers a range sum in $O(1)$ but pays $O(n)$ per update; a plain array updates in $O(1)$ but pays $O(n)$ per range sum. Fenwick and segment trees give us _both_ in $O(\\log n)$. The Fenwick (binary indexed) tree is a tiny array keyed by the low bit; the segment tree is a general balanced tree over canonical ranges that handles any associative aggregate and, with lazy propagation, range updates too.",{"module":121,"moduleNumber":102,"slug":122,"lessons":123},"Sequences & Strings","sequences",[124,130,135,141,147],{"title":125,"path":126,"lessonNumber":6,"topics":127,"summary":129},"Two Pointers, Sliding Windows & Prefix Sums","\u002Falgorithms\u002Fsequences\u002Ftwo-pointers-and-windows",[128],"Array Techniques","A family of array idioms that collapse an obvious $O(n^2)$ scan into a single $O(n)$ pass by maintaining an invariant as indices move. We meet two pointers (converging on a sorted array, and a fast\u002Fslow pair for in-place rewriting), the sliding window (fixed and variable size, amortized $O(n)$), and prefix sums, which answer any range-sum in $O(1)$ and count subarrays summing to $k$.",{"title":131,"path":132,"lessonNumber":18,"topics":133,"summary":134},"Monotonic Stacks & Queues","\u002Falgorithms\u002Fsequences\u002Fmonotonic-stacks",[128],"A **monotonic stack** keeps its contents sorted by popping every element that would break the order before each push — turning a family of \"previous\u002Fnext greater (or smaller) element\" questions into a single $O(n)$ scan. We derive the next-greater-element routine and its amortized analysis, fuse two such scans to measure the **largest rectangle in a histogram** in linear time, and extend the idea to a **monotonic deque** that streams the **sliding-window maximum** in $O(n)$.",{"title":136,"path":137,"lessonNumber":24,"topics":138,"summary":140},"Binary Search on the Answer","\u002Falgorithms\u002Fsequences\u002Fbinary-search-on-the-answer",[139],"Searching","Binary search is not really about arrays — it is about locating the boundary of a **monotone predicate** $p(x)$ in $O(\\log(\\text{range}))$ probes. We first pin down the half-open `while (lo \u003C hi)` template for $\\textsc{lower\\_bound}$ and $\\textsc{upper\\_bound}$, then generalize to \"binary search on the answer\": whenever feasibility is monotone in a numeric parameter, we binary search the parameter itself, calling a feasibility check at each step.",{"title":142,"path":143,"lessonNumber":73,"topics":144,"summary":146},"String Matching: Rabin–Karp, KMP & Z","\u002Falgorithms\u002Fsequences\u002Fstring-matching",[145],"Strings","Given a text $T$ of length $n$ and a pattern $P$ of length $m$, find every occurrence of $P$ in $T$. The naive scan costs $O(nm)$; we beat it three ways. Rabin–Karp uses a **rolling hash** to test alignments in $O(1)$ amortised each, with expected $O(n+m)$. KMP precomputes a **failure function** so the scan never re-reads a text character, for worst-case $O(n+m)$. The **Z-function** gives the same bound from a different angle and converts freely to KMP's table.",{"title":148,"path":149,"lessonNumber":102,"topics":150,"summary":151},"Tries & Prefix Trees","\u002Falgorithms\u002Fsequences\u002Ftries",[145],"A **trie** stores a set of strings in a tree keyed by _characters_, so that insert, search, and prefix-test all run in $O(L)$ time — the length of the key, _independent of how many keys are stored_. Shared prefixes are stored once, which makes tries the natural structure for autocomplete, wildcard dictionaries, board word-search, and — over the alphabet $\\{0,1\\}$ — the maximum-XOR-pair problem.",{"module":153,"moduleNumber":108,"slug":154,"lessons":155},"Graphs","graphs",[156,163,168,173,178,183,188,193,199],{"title":157,"path":158,"lessonNumber":6,"topics":159,"summary":162},"Graph Representations and Traversal","\u002Falgorithms\u002Fgraphs\u002Frepresentations-and-traversal",[160,161],"Graph Representations","Graph Traversal","A graph captures _relationships_ — who connects to whom. We fix the vocabulary, weigh the two standard representations (adjacency list versus matrix), then meet the two explorations you'll use constantly: breadth-first search, which finds shortest paths by number of edges, and depth-first search, whose discovery and finish times reveal a graph's hidden structure. Both run in $O(V + E)$.",{"title":164,"path":165,"lessonNumber":18,"topics":166,"summary":167},"Topological Sort and Strong Connectivity","\u002Falgorithms\u002Fgraphs\u002Ftopological-sort-and-scc",[161],"Directed acyclic graphs model dependencies: tasks that must precede other tasks. A _topological order_ lays such a graph out in a line so every edge points forward, and depth-first finish times hand it to us almost for free. We then ask the harder question for graphs _with_ cycles: which vertices can reach each other? The answer is the strongly connected components, found by a two-pass DFS.",{"title":169,"path":170,"lessonNumber":24,"topics":171,"summary":172},"Minimum Spanning Trees","\u002Falgorithms\u002Fgraphs\u002Fminimum-spanning-trees",[169],"Given a weighted network, how do we connect everything as cheaply as possible? The answer is a minimum spanning tree. One lemma, the cut property, justifies _every_ correct MST algorithm, and from it two famous greedy methods fall out: Kruskal's, which grows a forest edge by edge with a union-find structure, and Prim's, which grows a single tree using a priority queue.",{"title":174,"path":175,"lessonNumber":73,"topics":176,"summary":177},"Shortest Paths","\u002Falgorithms\u002Fgraphs\u002Fshortest-paths",[174],"Finding the cheapest route through a weighted network is one of the most-used algorithms in computing. A single operation — _relaxation_ — underlies them all. Dijkstra's algorithm solves the non-negative case greedily; Bellman-Ford handles negative edges and detects negative cycles; and Floyd-Warshall finds the shortest path between _every_ pair of vertices.",{"title":179,"path":180,"lessonNumber":102,"topics":181,"summary":182},"Network Flow","\u002Falgorithms\u002Fgraphs\u002Fnetwork-flow",[179],"How much can flow through a network from source to sink? Max-flow is a surprisingly general model — once you see a problem as flow, a whole toolbox opens up. We build flow networks, find maximum flows by repeatedly pushing along augmenting paths in the residual graph, prove the max-flow min-cut theorem, and watch bipartite matching fall out as a special case.",{"title":184,"path":185,"lessonNumber":108,"topics":186,"summary":187},"Bridges & Articulation Points","\u002Falgorithms\u002Fgraphs\u002Fbridges-and-articulation-points",[153],"A **bridge** is an edge whose removal disconnects the graph; an **articulation point** is a vertex whose removal does. Both are single points of failure in a network. A single depth-first search computes discovery times and **low-links**, and two local criteria — $low[v] > disc[u]$ for bridges, $low[v] \\ge disc[u]$ for cut vertices — find them all in $O(V+E)$.",{"title":189,"path":190,"lessonNumber":116,"topics":191,"summary":192},"Lowest Common Ancestor & Binary Lifting","\u002Falgorithms\u002Fgraphs\u002Flowest-common-ancestor",[153],"Given a rooted tree, the lowest common ancestor of $u$ and $v$ is the deepest node that is an ancestor of both. A naive walk answers one query in $O(h)$; **binary lifting** precomputes the $2^k$-th ancestor of every node in $O(n\\log n)$, then answers $k$-th-ancestor and LCA queries in $O(\\log n)$ each. We derive both jumps, apply them to tree distance, and compare against the Euler-tour + RMQ and Tarjan offline alternatives.",{"title":194,"path":195,"lessonNumber":196,"topics":197,"summary":198},"2-SAT via Implication Graphs","\u002Falgorithms\u002Fgraphs\u002Ftwo-sat",8,[153],"A boolean formula whose every clause has exactly two literals can be solved in _linear_ time — even though its three-literal cousin is NP-complete. The trick is to read each clause as a pair of implications, build a directed graph on the $2n$ literals, and ask a question we already know how to answer: which literals share a strongly connected component? The formula is satisfiable iff no variable lands in the same SCC as its own negation, and the SCCs' topological order hands us a satisfying assignment for free.",{"title":200,"path":201,"lessonNumber":202,"topics":203,"summary":204},"Eulerian Tours","\u002Falgorithms\u002Fgraphs\u002Feulerian-tours",9,[153],"An **Eulerian tour** uses every _edge_ of a graph exactly once. We give the exact parity and balance conditions under which one exists (even degree for undirected graphs, in-degree equal to out-degree for directed) and Hierholzer's $O(E)$ algorithm that constructs one by splicing closed sub-tours. We contrast this sharply with the **Hamiltonian** problem (visit every _vertex_ once), which is NP-complete: visiting edges is easy, visiting vertices is hard.",{"module":206,"moduleNumber":116,"slug":207,"lessons":208},"Greedy Algorithms","greedy",[209,214,220,225],{"title":210,"path":211,"lessonNumber":6,"topics":212,"summary":213},"The Greedy Method","\u002Falgorithms\u002Fgreedy\u002Fthe-greedy-method",[206],"A greedy algorithm builds a solution one locally-best choice at a time and never looks back. We pin down the two properties that make this work — the greedy-choice property and optimal substructure — prove the canonical activity-selection algorithm correct with an exchange argument, watch greedy fail spectacularly on the 0\u002F1 knapsack, and glimpse matroids as the theory that says exactly when greed is good.",{"title":215,"path":216,"lessonNumber":18,"topics":217,"summary":219},"Scheduling & Interval Partitioning","\u002Falgorithms\u002Fgreedy\u002Fscheduling-and-intervals",[218],"Greedy","Three classic scheduling problems all yield to greedy algorithms — and all three turn on a single design decision: which key to sort by. Interval scheduling sorts by **finish** time to pack the most compatible jobs; interval partitioning sorts by **start** time and proves the rooms needed equal the maximum overlap **depth**; minimizing maximum lateness sorts by **deadline** and is justified by an adjacent-swap exchange argument.",{"title":221,"path":222,"lessonNumber":24,"topics":223,"summary":224},"Huffman Codes","\u002Falgorithms\u002Fgreedy\u002Fhuffman-codes",[206],"Huffman coding is the greedy method's most beautiful application: it builds a provably optimal prefix-free binary code by repeatedly merging the two least frequent symbols. We develop prefix-free codes as binary trees, give the algorithm with a priority queue, build a Huffman tree from example frequencies, prove optimality with the same greedy-choice-plus-substructure argument, and pin the running time at $O(n\\log n)$.",{"title":226,"path":227,"lessonNumber":73,"topics":228,"summary":229},"Matroids & Exchange Arguments","\u002Falgorithms\u002Fgreedy\u002Fmatroids",[218],"The capstone of the greedy module: _why_ and _when_ a greedy algorithm is provably optimal. We recap the two correctness templates — **greedy-stays-ahead** and the **exchange argument** — then meet the **matroid** $M=(S,\\mathcal{I})$, an abstraction whose **exchange property** is exactly the structure greedy needs. The matroid–greedy theorem says sorting by weight and taking what stays independent yields a maximum-weight basis _if and only if_ the structure is a matroid. Kruskal's MST is the canonical instance; 0\u002F1 knapsack and TSP are the canonical failures.",{"module":231,"moduleNumber":196,"slug":232,"lessons":233},"Dynamic Programming","dynamic-programming",[234,239,245,250,255,260,265,270,275,280],{"title":235,"path":236,"lessonNumber":6,"topics":237,"summary":238},"Principles of Dynamic Programming","\u002Falgorithms\u002Fdynamic-programming\u002Fprinciples",[231,26],"Dynamic programming is recursion with memory: when a recursive solution re-solves the same subproblems again and again, we solve each one once and store the answer. We pin down the two structural conditions that make this work — overlapping subproblems and optimal substructure — contrast top-down memoization with bottom-up tabulation, and distil the whole method into a five-step recipe.",{"title":240,"path":241,"lessonNumber":18,"topics":242,"summary":244},"Sequence Alignment & LCS","\u002Falgorithms\u002Fdynamic-programming\u002Fsequence-dp",[231,243],"String Structures","Two strings can be compared by asking how much of one survives inside the other. The longest common subsequence (LCS) and edit distance are the two classic answers, and they are the _same_ dynamic program wearing different costs. We derive the LCS recurrence by examining the last characters, fill a worked DP table, reconstruct the subsequence, and then show edit distance as the identical $\\Theta(mn)$ pattern.",{"title":246,"path":247,"lessonNumber":24,"topics":248,"summary":249},"Longest Increasing Subsequence","\u002Falgorithms\u002Fdynamic-programming\u002Flongest-increasing-subsequence",[231],"Given a sequence of numbers, how long is its longest strictly increasing subsequence? A first dynamic program indexes subproblems by the element each subsequence _ends at_, giving an $O(n^2)$ solution with parent-pointer reconstruction. A sharper idea, the patience-sorting _tails_ array searched by binary search, drops the time to $O(n\\log n)$. We then fold in the variants: non-decreasing, counting, Russian-doll envelopes, and bitonic.",{"title":251,"path":252,"lessonNumber":73,"topics":253,"summary":254},"Knapsack & Subset Problems","\u002Falgorithms\u002Fdynamic-programming\u002Fknapsack",[231],"We start from $\\textsc{Subset-sum}$ — does some sublist hit a target $t$? — and its include\u002Fexclude recurrence over a boolean table $A(i, u)$, then bolt on values to get 0\u002F1 knapsack as the same machine with $\\lor$ promoted to $\\max$. We fill both tables, recover the chosen items, and confront the surprise that the $\\Theta(nt)$ running time is only _pseudo-polynomial_ — exponential in the bit length $b$, and unimprovable unless $\\mathrm{P}=\\mathrm{NP}$ since subset-sum is $\\textsc{NP-complete}$. The fractional variant reveals the sharp line between greedy and dynamic programming.",{"title":256,"path":257,"lessonNumber":102,"topics":258,"summary":259},"Coin Change & Unbounded Knapsack","\u002Falgorithms\u002Fdynamic-programming\u002Fcoin-change-and-unbounded",[231],"The previous lesson let each item be taken at most once. Drop that cap — items may be reused _any number of times_ — and the 0\u002F1 knapsack collapses from a two-dimensional table to a one-dimensional one, because there is no longer a prefix of \"already-used\" items to track. We meet **unbounded knapsack**, then its most famous instance, **coin change**: the minimum-coins recurrence $C[a] = 1 + \\min_c C[a-c]$, and the counting variant where the _order of the loops_ decides whether you count unordered combinations or ordered sequences — the classic bug. Greed fails in general but works for canonical coin systems.",{"title":261,"path":262,"lessonNumber":108,"topics":263,"summary":264},"Interval DP","\u002Falgorithms\u002Fdynamic-programming\u002Finterval-dp",[231],"Many problems ask for the best way to combine a contiguous range of items, and the answer is a dynamic program over subintervals $[i,j]$ that chooses a split point $k$. We derive the pattern from matrix-chain multiplication — parenthesising a product to minimize scalar multiplications in $O(n^3)$ — distil it into a reusable template filled by increasing interval length, and then meet its sharpest variant: the \"last operation\" trick behind Burst Balloons and cutting a stick, where fixing the _last_ move (not the first) makes the two sides independent.",{"title":266,"path":267,"lessonNumber":116,"topics":268,"summary":269},"Dynamic Programming on Trees","\u002Falgorithms\u002Fdynamic-programming\u002Ftree-dp",[231],"When the subproblems of a dynamic program are _rooted subtrees_, a single post-order DFS solves the whole thing in $O(n)$: each node combines the already-computed answers of its children. We meet the archetype — maximum-weight independent set on a tree — then the \"path through a node\" pattern behind tree diameter and maximum path sum, and finally **rerooting**, which computes a per-node answer for _every_ node as root in $O(n)$ with two passes.",{"title":271,"path":272,"lessonNumber":196,"topics":273,"summary":274},"Bitmask DP","\u002Falgorithms\u002Fdynamic-programming\u002Fbitmask-dp",[231],"When a subproblem depends not on an index or a prefix but on _which subset_ of a small ground set has been used, we can encode that subset as the bits of an integer and index a DP table by it. With $n \\le \\sim 20$ the $2^n$ subsets fit in a table, turning $\\Theta(n!)$ brute force into $O(2^n \\cdot \\text{poly}(n))$. We meet the bit tricks, the Held–Karp TSP archetype, assignment by mask, subset-sum partitioning, and submask enumeration with its $3^n$ bound.",{"title":276,"path":277,"lessonNumber":202,"topics":278,"summary":279},"DP Optimizations","\u002Falgorithms\u002Fdynamic-programming\u002Fdp-optimizations",[231],"A correct DP recurrence is only half the battle; its naive evaluation is often a factor of $n$ slower than necessary. This capstone surveys five techniques, monotonic-queue, the convex hull trick, divide-and-conquer optimization, Knuth's optimization, and SOS DP, that each exploit _structure in the transition_ (a sliding window, linear costs, monotone optimal splits, the quadrangle inequality, or subset lattices) to shave an $O(n)$, $O(\\log n)$, or worse factor off the running time.",{"title":281,"path":282,"lessonNumber":283,"topics":284,"summary":285},"Dynamic Programming on Graphs","\u002Falgorithms\u002Fdynamic-programming\u002Fdp-on-graphs",10,[231],"Many graph algorithms are dynamic programs in disguise: the subproblem is the _best value reachable under a restricted resource_ — intermediate vertices allowed, edges allowed, or a topological prefix — and edge _relaxation_ is the DP transition. We frame Floyd–Warshall as the archetype ($O(V^3)$ all-pairs shortest paths), Bellman–Ford as a DP over path length (the at-most-$K$-stops variant), DAG-DP in topological order ($O(V+E)$), and Warshall's transitive closure as the boolean analog.",{"module":287,"moduleNumber":202,"slug":288,"lessons":289},"Backtracking & Search","backtracking",[290,296,301],{"title":291,"path":292,"lessonNumber":6,"topics":293,"summary":295},"Backtracking: Subsets, Permutations & Combinations","\u002Falgorithms\u002Fbacktracking\u002Fbacktracking-fundamentals",[294],"Backtracking","Backtracking builds a solution one choice at a time and abandons a partial solution the moment it cannot be completed, exploring a state-space tree by depth-first search. We meet the universal choose\u002Fexplore\u002Fun-choose template, derive the canonical enumerations — subsets ($2^n$), permutations ($n!$), and combinations ($\\binom{n}{k}$) — handle duplicate elements by skipping equal siblings, and see how pruning turns an exponential search into a tractable one.",{"title":297,"path":298,"lessonNumber":18,"topics":299,"summary":300},"Constraint Search: N-Queens & Sudoku","\u002Falgorithms\u002Fbacktracking\u002Fconstraint-search",[294],"Many hard puzzles are **constraint satisfaction problems**: assign each variable a value from its domain so that every constraint holds. Backtracking solves them by assigning variables one at a time and rejecting a partial assignment the instant a constraint breaks. We make the rejection cheap — $O(1)$ conflict checks for N-Queens via column and diagonal sets — and prune harder with **forward checking**, **MRV** ordering, and **constraint propagation**, which is what lets an exponential search actually finish.",{"title":302,"path":303,"lessonNumber":24,"topics":304,"summary":305},"Branch & Bound and Meet in the Middle","\u002Falgorithms\u002Fbacktracking\u002Fbranch-and-bound",[294],"Plain backtracking prunes a search tree by _feasibility_; for _optimization_ problems we can prune far more aggressively by _value_. **Branch and bound** keeps the best complete solution found so far and discards any partial solution whose optimistic bound cannot beat it. **Meet in the middle** splits the instance in two, enumerates each half, and recombines by binary search — turning $2^n$ into $O(2^{n\u002F2}\\,n)$ and pushing exact search out to $n \\approx 40$.",{"module":307,"moduleNumber":283,"slug":308,"lessons":309},"Mathematical Algorithms","mathematical-algorithms",[310,316,321,326],{"title":311,"path":312,"lessonNumber":6,"topics":313,"summary":315},"Number Theory: GCD & Modular Arithmetic","\u002Falgorithms\u002Fmathematical-algorithms\u002Fnumber-theory-basics",[314],"Number Theory","This lesson opens the mathematical-algorithms module with the bedrock of computational number theory. We prove Euclid's recurrence $\\gcd(a,b)=\\gcd(b,\\,a\\bmod b)$ and its $O(\\log\\min(a,b))$ running time, extend it to recover Bézout coefficients $x,y$ with $ax+by=\\gcd(a,b)$, and build modular arithmetic on residue classes — including when a modular inverse $a^{-1}\\bmod m$ exists and how to compute it.",{"title":317,"path":318,"lessonNumber":18,"topics":319,"summary":320},"Modular Exponentiation & Primality","\u002Falgorithms\u002Fmathematical-algorithms\u002Fmodular-exponentiation-and-primality",[314],"Computing $a^n \\bmod m$ naively costs $n$ multiplications; **repeated squaring** does it in $O(\\log n)$ by reading the bits of the exponent. We use this routine to state **Fermat's little theorem** (and the modular inverse it gives), then to test primality — trial division, the probabilistic **Fermat** and **Miller–Rabin** tests, and the deterministic witness set that settles primality for every 64-bit number.",{"title":322,"path":323,"lessonNumber":24,"topics":324,"summary":325},"Sieves & Factorization","\u002Falgorithms\u002Fmathematical-algorithms\u002Fsieve-and-factorization",[314],"The previous lesson tested one number for primality; here we ask for _all_ primes up to $n$ at once. The **sieve of Eratosthenes** cross-cuts composites in $O(n\\log\\log n)$, and a **linear sieve** does it in $O(n)$ while recording each number's **smallest prime factor**, which then factors any $x \\le n$ in $O(\\log x)$. From a factorization $x = \\prod p_i^{e_i}$ the multiplicative functions $\\tau$, $\\sigma$, and Euler's totient $\\varphi$ fall out immediately.",{"title":327,"path":328,"lessonNumber":73,"topics":329,"summary":330},"Combinatorics & Counting","\u002Falgorithms\u002Fmathematical-algorithms\u002Fcombinatorics",[314],"Counting is the arithmetic of finite sets. We build up from permutations $n!$ and combinations $\\binom{n}{k}$, prove Pascal's rule by a bijection, and count multisets with stars and bars. The practical core is computing $\\binom{n}{k}\\bmod p$ in $O(1)$ from precomputed factorials and inverse factorials. We close with inclusion–exclusion and the Chinese Remainder Theorem, both of which lean on the modular inverse from the previous lesson.",{"module":332,"moduleNumber":333,"slug":334,"lessons":335},"Computational Geometry",11,"computational-geometry",[336,342,347],{"title":337,"path":338,"lessonNumber":6,"topics":339,"summary":341},"Geometric Primitives & Orientation","\u002Falgorithms\u002Fcomputational-geometry\u002Fgeometric-primitives",[340],"Geometry","Computational geometry is built on a single reliable primitive — the **orientation test**, a sign of a cross product that tells whether three points turn left, right, or lie collinear. From points-as-vectors and the dot and cross products we derive orientation, segment intersection, the shoelace area formula, and point-in-polygon tests, keeping all arithmetic **exact and integer** so that no floating-point rounding can corrupt a sign.",{"title":343,"path":344,"lessonNumber":18,"topics":345,"summary":346},"Convex Hull","\u002Falgorithms\u002Fcomputational-geometry\u002Fconvex-hull",[340],"The convex hull is the smallest convex polygon enclosing a point set — the rubber band snapped around the nails. We build it with Andrew's monotone chain, sorting by $(x,y)$ and sweeping a lower and upper hull while popping any non-left turn via the orientation primitive, in $O(n\\log n)$. A reduction from sorting shows that bound is optimal, and the hull unlocks diameter, smallest enclosing rectangle, and more through rotating calipers.",{"title":348,"path":349,"lessonNumber":24,"topics":350,"summary":351},"Sweep-Line Algorithms","\u002Falgorithms\u002Fcomputational-geometry\u002Fsweep-line",[340],"The plane-sweep paradigm turns a static $2$-D geometry problem into a dynamic $1$-D ordered-set problem: a vertical line sweeps left to right, stopping at an $x$-sorted **event queue** while a balanced-BST **status structure** tracks the objects it currently crosses, ordered by $y$. We derive Bentley–Ottmann segment intersection in $O((n+k)\\log n)$, recover closest-pair in $O(n\\log n)$, and reduce skyline, rectangle-area, and overlap problems to $\\pm1$ event sweeps.",{"module":353,"moduleNumber":354,"slug":355,"lessons":356},"Intractability",12,"intractability",[357,363,367],{"title":358,"path":359,"lessonNumber":6,"topics":360,"summary":362},"P, NP, and Reductions","\u002Falgorithms\u002Fintractability\u002Fp-np-reductions",[361],"NP-Completeness","Most problems we have met so far have fast algorithms. A vast and important family seemingly does not. This lesson builds the vocabulary for that divide: decision problems, the class $\\mathsf{P}$ of problems we can solve quickly, the class $\\mathsf{NP}$ of problems whose solutions we can _check_ quickly, and polynomial-time reductions, the tool that lets us compare the difficulty of two problems without solving either.",{"title":361,"path":364,"lessonNumber":18,"topics":365,"summary":366},"\u002Falgorithms\u002Fintractability\u002Fnp-completeness",[361],"Some problems in $\\mathsf{NP}$ are universally hardest: every other problem in $\\mathsf{NP}$ reduces to them. This lesson defines $\\mathsf{NP}$-hard and $\\mathsf{NP}$-complete, states the Cook–Levin theorem that anchors the whole edifice on **SAT**, walks the web of reductions that grows from it, and gives the four-step recipe for proving a brand-new problem $\\mathsf{NP}$-complete.",{"title":368,"path":369,"lessonNumber":24,"topics":370,"summary":373},"Coping with NP-Hardness","\u002Falgorithms\u002Fintractability\u002Fcoping-with-hardness",[371,372],"Approximation","Heuristics","Proving a problem $\\mathsf{NP}$-hard is the beginning, not the end. The world still needs answers. This lesson surveys the four honest responses to hardness: approximation algorithms with a provable ratio (worked through a 2-approximation for vertex cover), heuristics and local search, exact exponential methods like branch and bound, and exploiting special structure in the instances you actually face.",{"id":375,"title":361,"blurb":376,"body":377,"description":8323,"extension":8324,"meta":8325,"module":353,"navigation":8327,"path":364,"practice":8328,"rawbody":8343,"readingTime":8344,"seo":8349,"sources":8350,"status":8357,"stem":8358,"summary":366,"topics":8359,"__hash__":8360},"course\u002F01.algorithms\u002F12.intractability\u002F02.np-completeness.md","",{"type":378,"value":379,"toc":8313},"minimark",[380,464,469,680,970,1118,1325,1328,2198,2685,2729,2733,2759,2768,2993,3022,3523,4005,4027,4031,4526,4532,4706,4873,4878,5078,5649,5908,6051,6270,6680,6684,6762,7379,7465,7536,7540,8015],[381,382,383,384,412,413,417,418,434,435,453,454,458,459,463],"p",{},"The previous lesson left us with a claim: inside ",[385,386,389],"span",{"className":387},[388],"katex",[385,390,394],{"className":391,"ariaHidden":393},[392],"katex-html","true",[385,395,398,403],{"className":396},[397],"base",[385,399],{"className":400,"style":402},[401],"strut","height:0.6944em;",[385,404,407],{"className":405},[406],"mord",[385,408,411],{"className":409},[406,410],"mathsf","NP","\nthere are problems that are ",[414,415,416],"em",{},"universally hardest",", and they hold the key to the\n",[385,419,421],{"className":420},[388],[385,422,424],{"className":423,"ariaHidden":393},[392],[385,425,427,430],{"className":426},[397],[385,428],{"className":429,"style":402},[401],[385,431,433],{"className":432},[406,410],"P"," versus ",[385,436,438],{"className":437},[388],[385,439,441],{"className":440,"ariaHidden":393},[392],[385,442,444,447],{"className":443},[397],[385,445],{"className":446,"style":402},[401],[385,448,450],{"className":449},[406],[385,451,411],{"className":452},[406,410]," question. This lesson makes that claim\nprecise. We define what it means to be hardest, identify the first such problem\n(",[455,456,457],"strong",{},"SAT",", via the Cook–Levin theorem), and then show how a single seed problem\nsprouts an entire forest of equally hard problems through ",[460,461,462],"a",{"href":359},"reductions",". Finally\nwe distill the whole enterprise into a four-step recipe you can apply to a\nproblem you have never seen before.",[465,466,468],"h2",{"id":467},"np-hard-and-np-complete","NP-hard and NP-complete",[381,470,471,472,569,570,605,606,621,622,625,626,644,645,660,661,679],{},"Recall that ",[385,473,475],{"className":474},[388],[385,476,478,557],{"className":477,"ariaHidden":393},[392],[385,479,481,485,490,495,554],{"className":480},[397],[385,482],{"className":483,"style":484},[401],"height:0.8333em;vertical-align:-0.15em;",[385,486,489],{"className":487},[406,488],"mathnormal","A",[385,491],{"className":492,"style":494},[493],"mspace","margin-right:0.2778em;",[385,496,499,503],{"className":497},[498],"mrel",[385,500,502],{"className":501},[498],"≤",[385,504,507],{"className":505},[506],"msupsub",[385,508,512,545],{"className":509},[510,511],"vlist-t","vlist-t2",[385,513,516,540],{"className":514},[515],"vlist-r",[385,517,521],{"className":518,"style":520},[519],"vlist","height:0.3283em;",[385,522,524,529],{"style":523},"top:-2.55em;margin-left:0em;margin-right:0.05em;",[385,525],{"className":526,"style":528},[527],"pstrut","height:2.7em;",[385,530,536],{"className":531},[532,533,534,535],"sizing","reset-size6","size3","mtight",[385,537,433],{"className":538,"style":539},[406,488,535],"margin-right:0.1389em;",[385,541,544],{"className":542},[543],"vlist-s","​",[385,546,548],{"className":547},[515],[385,549,552],{"className":550,"style":551},[519],"height:0.15em;",[385,553],{},[385,555],{"className":556,"style":494},[493],[385,558,560,564],{"className":559},[397],[385,561],{"className":562,"style":563},[401],"height:0.6833em;",[385,565,568],{"className":566,"style":567},[406,488],"margin-right:0.0502em;","B"," means ",[571,572,573,588,589,604],"q",{},[385,574,576],{"className":575},[388],[385,577,579],{"className":578,"ariaHidden":393},[392],[385,580,582,585],{"className":581},[397],[385,583],{"className":584,"style":563},[401],[385,586,489],{"className":587},[406,488]," is no harder than ",[385,590,592],{"className":591},[388],[385,593,595],{"className":594,"ariaHidden":393},[392],[385,596,598,601],{"className":597},[397],[385,599],{"className":600,"style":563},[401],[385,602,568],{"className":603,"style":567},[406,488],"."," Now imagine a\nproblem ",[385,607,609],{"className":608},[388],[385,610,612],{"className":611,"ariaHidden":393},[392],[385,613,615,618],{"className":614},[397],[385,616],{"className":617,"style":563},[401],[385,619,568],{"className":620,"style":567},[406,488]," that ",[414,623,624],{},"every"," problem in ",[385,627,629],{"className":628},[388],[385,630,632],{"className":631,"ariaHidden":393},[392],[385,633,635,638],{"className":634},[397],[385,636],{"className":637,"style":402},[401],[385,639,641],{"className":640},[406],[385,642,411],{"className":643},[406,410]," is no harder than. Such a ",[385,646,648],{"className":647},[388],[385,649,651],{"className":650,"ariaHidden":393},[392],[385,652,654,657],{"className":653},[397],[385,655],{"className":656,"style":563},[401],[385,658,568],{"className":659,"style":567},[406,488],"\nis at least as hard as everything in ",[385,662,664],{"className":663},[388],[385,665,667],{"className":666,"ariaHidden":393},[392],[385,668,670,673],{"className":669},[397],[385,671],{"className":672,"style":402},[401],[385,674,676],{"className":675},[406],[385,677,411],{"className":678},[406,410],", a ceiling on the whole\nclass.",[681,682,684,875],"callout",{"type":683},"definition",[381,685,686,727,728,743,744,762,763,833,834,836,837,604],{},[455,687,688,689,707,708,726],{},"Definition (",[385,690,692],{"className":691},[388],[385,693,695],{"className":694,"ariaHidden":393},[392],[385,696,698,701],{"className":697},[397],[385,699],{"className":700,"style":402},[401],[385,702,704],{"className":703},[406],[385,705,411],{"className":706},[406,410],"-hard and ",[385,709,711],{"className":710},[388],[385,712,714],{"className":713,"ariaHidden":393},[392],[385,715,717,720],{"className":716},[397],[385,718],{"className":719,"style":402},[401],[385,721,723],{"className":722},[406],[385,724,411],{"className":725},[406,410],"-complete)."," A problem ",[385,729,731],{"className":730},[388],[385,732,734],{"className":733,"ariaHidden":393},[392],[385,735,737,740],{"className":736},[397],[385,738],{"className":739,"style":563},[401],[385,741,568],{"className":742,"style":567},[406,488]," is ",[385,745,747],{"className":746},[388],[385,748,750],{"className":749,"ariaHidden":393},[392],[385,751,753,756],{"className":752},[397],[385,754],{"className":755,"style":402},[401],[385,757,759],{"className":758},[406],[385,760,411],{"className":761},[406,410],"-hard if ",[385,764,766],{"className":765},[388],[385,767,769,824],{"className":768,"ariaHidden":393},[392],[385,770,772,775,778,781,821],{"className":771},[397],[385,773],{"className":774,"style":484},[401],[385,776,489],{"className":777},[406,488],[385,779],{"className":780,"style":494},[493],[385,782,784,787],{"className":783},[498],[385,785,502],{"className":786},[498],[385,788,790],{"className":789},[506],[385,791,793,813],{"className":792},[510,511],[385,794,796,810],{"className":795},[515],[385,797,799],{"className":798,"style":520},[519],[385,800,801,804],{"style":523},[385,802],{"className":803,"style":528},[527],[385,805,807],{"className":806},[532,533,534,535],[385,808,433],{"className":809,"style":539},[406,488,535],[385,811,544],{"className":812},[543],[385,814,816],{"className":815},[515],[385,817,819],{"className":818,"style":551},[519],[385,820],{},[385,822],{"className":823,"style":494},[493],[385,825,827,830],{"className":826},[397],[385,828],{"className":829,"style":563},[401],[385,831,568],{"className":832,"style":567},[406,488]," for ",[414,835,624],{}," problem ",[385,838,840],{"className":839},[388],[385,841,843,863],{"className":842,"ariaHidden":393},[392],[385,844,846,850,853,856,860],{"className":845},[397],[385,847],{"className":848,"style":849},[401],"height:0.7224em;vertical-align:-0.0391em;",[385,851,489],{"className":852},[406,488],[385,854],{"className":855,"style":494},[493],[385,857,859],{"className":858},[498],"∈",[385,861],{"className":862,"style":494},[493],[385,864,866,869],{"className":865},[397],[385,867],{"className":868,"style":402},[401],[385,870,872],{"className":871},[406],[385,873,411],{"className":874},[406,410],[381,876,877,878,743,893,911,912,930,931,934,604],{},"A problem ",[385,879,881],{"className":880},[388],[385,882,884],{"className":883,"ariaHidden":393},[392],[385,885,887,890],{"className":886},[397],[385,888],{"className":889,"style":563},[401],[385,891,568],{"className":892,"style":567},[406,488],[385,894,896],{"className":895},[388],[385,897,899],{"className":898,"ariaHidden":393},[392],[385,900,902,905],{"className":901},[397],[385,903],{"className":904,"style":402},[401],[385,906,908],{"className":907},[406],[385,909,411],{"className":910},[406,410],"-complete if it is ",[385,913,915],{"className":914},[388],[385,916,918],{"className":917,"ariaHidden":393},[392],[385,919,921,924],{"className":920},[397],[385,922],{"className":923,"style":402},[401],[385,925,927],{"className":926},[406],[385,928,411],{"className":929},[406,410],"-hard ",[414,932,933],{},"and",[385,935,937],{"className":936},[388],[385,938,940,958],{"className":939,"ariaHidden":393},[392],[385,941,943,946,949,952,955],{"className":942},[397],[385,944],{"className":945,"style":849},[401],[385,947,568],{"className":948,"style":567},[406,488],[385,950],{"className":951,"style":494},[493],[385,953,859],{"className":954},[498],[385,956],{"className":957,"style":494},[493],[385,959,961,964],{"className":960},[397],[385,962],{"className":963,"style":402},[401],[385,965,967],{"className":966},[406],[385,968,411],{"className":969},[406,410],[381,971,972,973,991,992,995,996,1011,1012,1030,1031,1046,1047,1065,1066,1084,1085,1088,1089,1107,1108],{},"The distinction matters. ",[385,974,976],{"className":975},[388],[385,977,979],{"className":978,"ariaHidden":393},[392],[385,980,982,985],{"className":981},[397],[385,983],{"className":984,"style":402},[401],[385,986,988],{"className":987},[406],[385,989,411],{"className":990},[406,410],"-hardness is a pure ",[414,993,994],{},"lower bound",": ",[385,997,999],{"className":998},[388],[385,1000,1002],{"className":1001,"ariaHidden":393},[392],[385,1003,1005,1008],{"className":1004},[397],[385,1006],{"className":1007,"style":563},[401],[385,1009,568],{"className":1010,"style":567},[406,488]," is\nas hard as anything in ",[385,1013,1015],{"className":1014},[388],[385,1016,1018],{"className":1017,"ariaHidden":393},[392],[385,1019,1021,1024],{"className":1020},[397],[385,1022],{"className":1023,"style":402},[401],[385,1025,1027],{"className":1026},[406],[385,1028,411],{"className":1029},[406,410],", but ",[385,1032,1034],{"className":1033},[388],[385,1035,1037],{"className":1036,"ariaHidden":393},[392],[385,1038,1040,1043],{"className":1039},[397],[385,1041],{"className":1042,"style":563},[401],[385,1044,568],{"className":1045,"style":567},[406,488]," need not itself be in\n",[385,1048,1050],{"className":1049},[388],[385,1051,1053],{"className":1052,"ariaHidden":393},[392],[385,1054,1056,1059],{"className":1055},[397],[385,1057],{"className":1058,"style":402},[401],[385,1060,1062],{"className":1061},[406],[385,1063,411],{"className":1064},[406,410],"; it could be far harder, even undecidable. ",[385,1067,1069],{"className":1068},[388],[385,1070,1072],{"className":1071,"ariaHidden":393},[392],[385,1073,1075,1078],{"className":1074},[397],[385,1076],{"className":1077,"style":402},[401],[385,1079,1081],{"className":1080},[406],[385,1082,411],{"className":1083},[406,410],"-complete\nproblems are the ones that are hardest ",[414,1086,1087],{},"and still belong to"," ",[385,1090,1092],{"className":1091},[388],[385,1093,1095],{"className":1094,"ariaHidden":393},[392],[385,1096,1098,1101],{"className":1097},[397],[385,1099],{"className":1100,"style":402},[401],[385,1102,1104],{"className":1103},[406],[385,1105,411],{"className":1106},[406,410],":\nthey sit exactly at the frontier. They are the hardest\nproblems whose solutions we can still efficiently check.",[1109,1110,1111],"sup",{},[460,1112,1117],{"href":1113,"ariaDescribedBy":1114,"dataFootnoteRef":376,"id":1116},"#user-content-fn-clrs-nphard",[1115],"footnote-label","user-content-fnref-clrs-nphard","1",[1119,1120,1124,1226],"figure",{"className":1121},[1122,1123],"tikz-figure","tikz-diagram-rendered",[1125,1126,1131],"svg",{"xmlns":1127,"width":1128,"height":1129,"viewBox":1130},"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg","321.451","138.052","-75 -75 241.089 103.539",[1132,1133,1136,1142,1151,1154,1169,1174,1182,1210],"g",{"stroke":1134,"style":1135},"currentColor","stroke-miterlimit:10;stroke-width:.4",[1137,1138],"path",{"fill":1139,"d":1140,"style":1141},"none","M77.06-23.5c0-26.715-31.846-48.37-71.132-48.37-39.285 0-71.131 21.655-71.131 48.37s31.846 48.37 71.131 48.37c39.286 0 71.132-21.656 71.132-48.37Zm-71.132 0","stroke-width:.8",[1132,1143,1145],{"transform":1144},"translate(-46.063 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37.951-25.323Q38.324-25.376 38.628-25.376L38.628-25.629Q38.628-25.834 38.520-26.014Q38.412-26.193 38.231-26.296Q38.050-26.398 37.842-26.398Q37.435-26.398 37.199-26.306Q37.288-26.269 37.334-26.185Q37.380-26.101 37.380-25.999Q37.380-25.903 37.334-25.824Q37.288-25.746 37.208-25.701Q37.127-25.657 37.038-25.657Q36.888-25.657 36.787-25.754Q36.686-25.852 36.686-25.999Q36.686-26.621 37.842-26.621Q38.053-26.621 38.303-26.557Q38.553-26.494 38.754-26.375Q38.956-26.255 39.082-26.070Q39.209-25.886 39.209-25.643L39.209-24.067Q39.209-23.951 39.270-23.855Q39.332-23.760 39.445-23.760Q39.554-23.760 39.619-23.854Q39.684-23.948 39.684-24.067L39.684-24.515L39.950-24.515L39.950-24.067Q39.950-23.797 39.723-23.632Q39.496-23.466 39.216-23.466Q39.007-23.466 38.870-23.620Q38.734-23.773 38.710-23.989Q38.563-23.722 38.281-23.577Q37.999-23.432 37.674-23.432Q37.397-23.432 37.114-23.507Q36.830-23.582 36.637-23.761Q36.444-23.941 36.444-24.228M37.059-24.228Q37.059-24.054 37.160-23.924Q37.261-23.794 37.416-23.724Q37.572-23.654 37.736-23.654Q37.954-23.654 38.163-23.751Q38.371-23.849 38.500-24.030Q38.628-24.211 38.628-24.437L38.628-25.165Q38.303-25.165 37.937-25.074Q37.572-24.983 37.315-24.771Q37.059-24.560 37.059-24.228M41.174-23.500L40.907-23.500L40.907-27.608Q40.907-27.878 40.800-27.940Q40.692-28.001 40.381-28.001L40.381-28.282L41.461-28.357L41.461-26.187Q41.670-26.378 41.955-26.482Q42.240-26.586 42.538-26.586Q42.856-26.586 43.153-26.465Q43.450-26.344 43.673-26.128Q43.895-25.913 44.021-25.628Q44.148-25.342 44.148-25.011Q44.148-24.566 43.908-24.202Q43.669-23.838 43.276-23.635Q42.883-23.432 42.439-23.432Q42.244-23.432 42.054-23.488Q41.865-23.544 41.704-23.649Q41.543-23.753 41.403-23.914L41.174-23.500M41.489-25.845L41.489-24.228Q41.625-23.968 41.866-23.811Q42.107-23.654 42.384-23.654Q42.678-23.654 42.890-23.761Q43.102-23.869 43.235-24.061Q43.368-24.252 43.427-24.491Q43.485-24.730 43.485-25.011Q43.485-25.370 43.391-25.674Q43.297-25.978 43.069-26.171Q42.842-26.364 42.476-26.364Q42.176-26.364 41.909-26.228Q41.642-26.091 41.489-25.845M46.451-23.500L44.848-23.500L44.848-23.780Q45.074-23.780 45.223-23.814Q45.371-23.849 45.371-23.989L45.371-27.608Q45.371-27.878 45.264-27.940Q45.156-28.001 44.848-28.001L44.848-28.282L45.925-28.357L45.925-23.989Q45.925-23.852 46.075-23.816Q46.226-23.780 46.451-23.780L46.451-23.500M47.005-25.035Q47.005-25.356 47.130-25.645Q47.255-25.934 47.480-26.157Q47.706-26.381 48.001-26.501Q48.297-26.621 48.615-26.621Q48.943-26.621 49.205-26.521Q49.466-26.422 49.642-26.240Q49.818-26.057 49.912-25.799Q50.006-25.541 50.006-25.209Q50.006-25.117 49.924-25.096L47.668-25.096L47.668-25.035Q47.668-24.447 47.952-24.064Q48.236-23.681 48.803-23.681Q49.124-23.681 49.393-23.874Q49.661-24.067 49.750-24.382Q49.757-24.423 49.832-24.437L49.924-24.437Q50.006-24.413 50.006-24.341Q50.006-24.334 49.999-24.307Q49.886-23.910 49.516-23.671Q49.145-23.432 48.721-23.432Q48.283-23.432 47.884-23.640Q47.484-23.849 47.244-24.216Q47.005-24.583 47.005-25.035M47.675-25.305L49.490-25.305Q49.490-25.582 49.393-25.834Q49.295-26.087 49.097-26.243Q48.899-26.398 48.615-26.398Q48.338-26.398 48.125-26.240Q47.911-26.081 47.793-25.826Q47.675-25.571 47.675-25.305M50.915-21.743L50.847-21.743Q50.813-21.743 50.791-21.769Q50.768-21.794 50.768-21.829Q50.768-21.873 50.799-21.890Q51.155-22.194 51.404-22.584Q51.654-22.974 51.806-23.406Q51.958-23.838 52.028-24.307Q52.098-24.775 52.098-25.250Q52.098-25.729 52.028-26.195Q51.958-26.662 51.804-27.097Q51.650-27.533 51.399-27.921Q51.148-28.309 50.799-28.603Q50.768-28.620 50.768-28.665Q50.768-28.699 50.791-28.724Q50.813-28.750 50.847-28.750L50.915-28.750Q50.926-28.750 50.934-28.748Q50.943-28.747 50.953-28.743Q51.496-28.343 51.869-27.790Q52.241-27.236 52.423-26.590Q52.604-25.944 52.604-25.250Q52.604-24.549 52.423-23.902Q52.241-23.254 51.867-22.700Q51.493-22.146 50.953-21.750Q50.943-21.750 50.934-21.748Q50.926-21.747 50.915-21.743",[1149],[1227,1228,1231,1249,1250,1268,1269,1287,1288,1306,1307,604],"figcaption",{"className":1229},[1230],"tikz-cap",[385,1232,1234],{"className":1233},[388],[385,1235,1237],{"className":1236,"ariaHidden":393},[392],[385,1238,1240,1243],{"className":1239},[397],[385,1241],{"className":1242,"style":402},[401],[385,1244,1246],{"className":1245},[406],[385,1247,411],{"className":1248},[406,410],"-complete is the intersection of ",[385,1251,1253],{"className":1252},[388],[385,1254,1256],{"className":1255,"ariaHidden":393},[392],[385,1257,1259,1262],{"className":1258},[397],[385,1260],{"className":1261,"style":402},[401],[385,1263,1265],{"className":1264},[406],[385,1266,411],{"className":1267},[406,410]," and ",[385,1270,1272],{"className":1271},[388],[385,1273,1275],{"className":1274,"ariaHidden":393},[392],[385,1276,1278,1281],{"className":1277},[397],[385,1279],{"className":1280,"style":402},[401],[385,1282,1284],{"className":1283},[406],[385,1285,411],{"className":1286},[406,410],"-hard; some ",[385,1289,1291],{"className":1290},[388],[385,1292,1294],{"className":1293,"ariaHidden":393},[392],[385,1295,1297,1300],{"className":1296},[397],[385,1298],{"className":1299,"style":402},[401],[385,1301,1303],{"className":1302},[406],[385,1304,411],{"className":1305},[406,410],"-hard problems lie outside ",[385,1308,1310],{"className":1309},[388],[385,1311,1313],{"className":1312,"ariaHidden":393},[392],[385,1314,1316,1319],{"className":1315},[397],[385,1317],{"className":1318,"style":402},[401],[385,1320,1322],{"className":1321},[406],[385,1323,411],{"className":1324},[406,410],[381,1326,1327],{},"Two consequences make these definitions powerful.",[1329,1330,1331,1773],"ul",{},[1332,1333,1334,1356,1357,1372,1373,1391,1392,1407,1408,1088,1411,1447,1448,1518,1519,1534,1535,1550,1551,1654,1655,1088,1658,1676,1677,1753,1754,1772],"li",{},[455,1335,1336,1337,1355],{},"All ",[385,1338,1340],{"className":1339},[388],[385,1341,1343],{"className":1342,"ariaHidden":393},[392],[385,1344,1346,1349],{"className":1345},[397],[385,1347],{"className":1348,"style":402},[401],[385,1350,1352],{"className":1351},[406],[385,1353,411],{"className":1354},[406,410],"-complete problems stand or fall together."," Suppose ",[385,1358,1360],{"className":1359},[388],[385,1361,1363],{"className":1362,"ariaHidden":393},[392],[385,1364,1366,1369],{"className":1365},[397],[385,1367],{"className":1368,"style":563},[401],[385,1370,568],{"className":1371,"style":567},[406,488]," is\n",[385,1374,1376],{"className":1375},[388],[385,1377,1379],{"className":1378,"ariaHidden":393},[392],[385,1380,1382,1385],{"className":1381},[397],[385,1383],{"className":1384,"style":402},[401],[385,1386,1388],{"className":1387},[406],[385,1389,411],{"className":1390},[406,410],"-complete and someone finds a polynomial-time algorithm for ",[385,1393,1395],{"className":1394},[388],[385,1396,1398],{"className":1397,"ariaHidden":393},[392],[385,1399,1401,1404],{"className":1400},[397],[385,1402],{"className":1403,"style":563},[401],[385,1405,568],{"className":1406,"style":567},[406,488],".\nThen for ",[414,1409,1410],{},"any",[385,1412,1414],{"className":1413},[388],[385,1415,1417,1435],{"className":1416,"ariaHidden":393},[392],[385,1418,1420,1423,1426,1429,1432],{"className":1419},[397],[385,1421],{"className":1422,"style":849},[401],[385,1424,489],{"className":1425},[406,488],[385,1427],{"className":1428,"style":494},[493],[385,1430,859],{"className":1431},[498],[385,1433],{"className":1434,"style":494},[493],[385,1436,1438,1441],{"className":1437},[397],[385,1439],{"className":1440,"style":402},[401],[385,1442,1444],{"className":1443},[406],[385,1445,411],{"className":1446},[406,410]," we have ",[385,1449,1451],{"className":1450},[388],[385,1452,1454,1509],{"className":1453,"ariaHidden":393},[392],[385,1455,1457,1460,1463,1466,1506],{"className":1456},[397],[385,1458],{"className":1459,"style":484},[401],[385,1461,489],{"className":1462},[406,488],[385,1464],{"className":1465,"style":494},[493],[385,1467,1469,1472],{"className":1468},[498],[385,1470,502],{"className":1471},[498],[385,1473,1475],{"className":1474},[506],[385,1476,1478,1498],{"className":1477},[510,511],[385,1479,1481,1495],{"className":1480},[515],[385,1482,1484],{"className":1483,"style":520},[519],[385,1485,1486,1489],{"style":523},[385,1487],{"className":1488,"style":528},[527],[385,1490,1492],{"className":1491},[532,533,534,535],[385,1493,433],{"className":1494,"style":539},[406,488,535],[385,1496,544],{"className":1497},[543],[385,1499,1501],{"className":1500},[515],[385,1502,1504],{"className":1503,"style":551},[519],[385,1505],{},[385,1507],{"className":1508,"style":494},[493],[385,1510,1512,1515],{"className":1511},[397],[385,1513],{"className":1514,"style":563},[401],[385,1516,568],{"className":1517,"style":567},[406,488],", so ",[385,1520,1522],{"className":1521},[388],[385,1523,1525],{"className":1524,"ariaHidden":393},[392],[385,1526,1528,1531],{"className":1527},[397],[385,1529],{"className":1530,"style":563},[401],[385,1532,489],{"className":1533},[406,488]," is solvable in\npolynomial time too, every ",[385,1536,1538],{"className":1537},[388],[385,1539,1541],{"className":1540,"ariaHidden":393},[392],[385,1542,1544,1547],{"className":1543},[397],[385,1545],{"className":1546,"style":563},[401],[385,1548,489],{"className":1549},[406,488]," at once. Hence\n",[385,1552,1554],{"className":1553},[388],[385,1555,1557,1594,1620,1639],{"className":1556,"ariaHidden":393},[392],[385,1558,1560,1564,1572,1578,1585,1588,1591],{"className":1559},[397],[385,1561],{"className":1562,"style":1563},[401],"height:0.8889em;vertical-align:-0.1944em;",[385,1565,1568],{"className":1566},[406,1567],"text",[385,1569,1571],{"className":1570},[406],"any ",[385,1573,1575],{"className":1574},[406],[385,1576,411],{"className":1577},[406,410],[385,1579,1581],{"className":1580},[406,1567],[385,1582,1584],{"className":1583},[406],"-complete problem",[385,1586],{"className":1587,"style":494},[493],[385,1589,859],{"className":1590},[498],[385,1592],{"className":1593,"style":494},[493],[385,1595,1597,1601,1604,1607,1610,1614,1617],{"className":1596},[397],[385,1598],{"className":1599,"style":1600},[401],"height:0.7184em;vertical-align:-0.024em;",[385,1602,433],{"className":1603},[406,410],[385,1605],{"className":1606,"style":494},[493],[385,1608],{"className":1609,"style":494},[493],[385,1611,1613],{"className":1612},[498],"⟹",[385,1615],{"className":1616,"style":494},[493],[385,1618],{"className":1619,"style":494},[493],[385,1621,1623,1626,1629,1632,1636],{"className":1622},[397],[385,1624],{"className":1625,"style":402},[401],[385,1627,433],{"className":1628},[406,410],[385,1630],{"className":1631,"style":494},[493],[385,1633,1635],{"className":1634},[498],"=",[385,1637],{"className":1638,"style":494},[493],[385,1640,1642,1645,1651],{"className":1641},[397],[385,1643],{"className":1644,"style":402},[401],[385,1646,1648],{"className":1647},[406],[385,1649,411],{"className":1650},[406,410],[385,1652,604],{"className":1653},[406],"\nConversely, proving ",[414,1656,1657],{},"any single",[385,1659,1661],{"className":1660},[388],[385,1662,1664],{"className":1663,"ariaHidden":393},[392],[385,1665,1667,1670],{"className":1666},[397],[385,1668],{"className":1669,"style":402},[401],[385,1671,1673],{"className":1672},[406],[385,1674,411],{"className":1675},[406,410],"-complete problem requires\nsuper-polynomial time would prove ",[385,1678,1680],{"className":1679},[388],[385,1681,1683,1741],{"className":1682,"ariaHidden":393},[392],[385,1684,1686,1689,1692,1695,1738],{"className":1685},[397],[385,1687],{"className":1688,"style":1563},[401],[385,1690,433],{"className":1691},[406,410],[385,1693],{"className":1694,"style":494},[493],[385,1696,1698,1731,1735],{"className":1697},[498],[385,1699,1701],{"className":1700},[498],[385,1702,1705],{"className":1703},[406,1704],"vbox",[385,1706,1709],{"className":1707},[1708],"thinbox",[385,1710,1713,1716,1727],{"className":1711},[1712],"rlap",[385,1714],{"className":1715,"style":1563},[401],[385,1717,1720],{"className":1718},[1719],"inner",[385,1721,1723],{"className":1722},[406],[385,1724,1726],{"className":1725},[498],"",[385,1728],{"className":1729},[1730],"fix",[385,1732],{"className":1733},[493,1734],"nobreak",[385,1736,1635],{"className":1737},[498],[385,1739],{"className":1740,"style":494},[493],[385,1742,1744,1747],{"className":1743},[397],[385,1745],{"className":1746,"style":402},[401],[385,1748,1750],{"className":1749},[406],[385,1751,411],{"className":1752},[406,410],". In this sense\nthe thousands of known ",[385,1755,1757],{"className":1756},[388],[385,1758,1760],{"className":1759,"ariaHidden":393},[392],[385,1761,1763,1766],{"className":1762},[397],[385,1764],{"className":1765,"style":402},[401],[385,1767,1769],{"className":1768},[406],[385,1770,411],{"className":1771},[406,410],"-complete problems are a single problem\nin many guises.",[1332,1774,1775,1778,1779,743,1796,1814,1815,1887,1888,1924,1925,1372,1940,1958,1959,1995,1996,2121,2122,2175,2176,2179,2197],{},[455,1776,1777],{},"Transitivity bootstraps the class."," If ",[385,1780,1782],{"className":1781},[388],[385,1783,1785],{"className":1784,"ariaHidden":393},[392],[385,1786,1788,1791],{"className":1787},[397],[385,1789],{"className":1790,"style":563},[401],[385,1792,1795],{"className":1793,"style":1794},[406,488],"margin-right:0.0715em;","C",[385,1797,1799],{"className":1798},[388],[385,1800,1802],{"className":1801,"ariaHidden":393},[392],[385,1803,1805,1808],{"className":1804},[397],[385,1806],{"className":1807,"style":402},[401],[385,1809,1811],{"className":1810},[406],[385,1812,411],{"className":1813},[406,410],"-complete and we\nshow ",[385,1816,1818],{"className":1817},[388],[385,1819,1821,1876],{"className":1820,"ariaHidden":393},[392],[385,1822,1824,1827,1830,1833,1873],{"className":1823},[397],[385,1825],{"className":1826,"style":484},[401],[385,1828,1795],{"className":1829,"style":1794},[406,488],[385,1831],{"className":1832,"style":494},[493],[385,1834,1836,1839],{"className":1835},[498],[385,1837,502],{"className":1838},[498],[385,1840,1842],{"className":1841},[506],[385,1843,1845,1865],{"className":1844},[510,511],[385,1846,1848,1862],{"className":1847},[515],[385,1849,1851],{"className":1850,"style":520},[519],[385,1852,1853,1856],{"style":523},[385,1854],{"className":1855,"style":528},[527],[385,1857,1859],{"className":1858},[532,533,534,535],[385,1860,433],{"className":1861,"style":539},[406,488,535],[385,1863,544],{"className":1864},[543],[385,1866,1868],{"className":1867},[515],[385,1869,1871],{"className":1870,"style":551},[519],[385,1872],{},[385,1874],{"className":1875,"style":494},[493],[385,1877,1879,1882],{"className":1878},[397],[385,1880],{"className":1881,"style":563},[401],[385,1883,1886],{"className":1884,"style":1885},[406,488],"margin-right:0.0278em;","D"," for some ",[385,1889,1891],{"className":1890},[388],[385,1892,1894,1912],{"className":1893,"ariaHidden":393},[392],[385,1895,1897,1900,1903,1906,1909],{"className":1896},[397],[385,1898],{"className":1899,"style":849},[401],[385,1901,1886],{"className":1902,"style":1885},[406,488],[385,1904],{"className":1905,"style":494},[493],[385,1907,859],{"className":1908},[498],[385,1910],{"className":1911,"style":494},[493],[385,1913,1915,1918],{"className":1914},[397],[385,1916],{"className":1917,"style":402},[401],[385,1919,1921],{"className":1920},[406],[385,1922,411],{"className":1923},[406,410],", then 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180.987-22.864Q181.308-22.864 181.576-23.057Q181.845-23.250 181.933-23.565Q181.940-23.606 182.015-23.620L182.108-23.620Q182.190-23.596 182.190-23.524Q182.190-23.517 182.183-23.490Q182.070-23.093 181.699-22.854Q181.328-22.615 180.905-22.615Q180.467-22.615 180.067-22.823Q179.667-23.032 179.428-23.399Q179.189-23.766 179.189-24.218M179.859-24.488L181.674-24.488Q181.674-24.765 181.576-25.017Q181.479-25.270 181.281-25.426Q181.082-25.581 180.799-25.581Q180.522-25.581 180.308-25.423Q180.095-25.264 179.977-25.009Q179.859-24.754 179.859-24.488",[1149],[1227,2542,2544,2545,2581,2582,2597,2598,2668,2669,2684],{"className":2543},[1230],"Transitivity bootstrap: every ",[385,2546,2548],{"className":2547},[388],[385,2549,2551,2569],{"className":2550,"ariaHidden":393},[392],[385,2552,2554,2557,2560,2563,2566],{"className":2553},[397],[385,2555],{"className":2556,"style":849},[401],[385,2558,489],{"className":2559},[406,488],[385,2561],{"className":2562,"style":494},[493],[385,2564,859],{"className":2565},[498],[385,2567],{"className":2568,"style":494},[493],[385,2570,2572,2575],{"className":2571},[397],[385,2573],{"className":2574,"style":402},[401],[385,2576,2578],{"className":2577},[406],[385,2579,411],{"className":2580},[406,410]," already reduces to the known-complete ",[385,2583,2585],{"className":2584},[388],[385,2586,2588],{"className":2587,"ariaHidden":393},[392],[385,2589,2591,2594],{"className":2590},[397],[385,2592],{"className":2593,"style":563},[401],[385,2595,1795],{"className":2596,"style":1794},[406,488],"; one new reduction ",[385,2599,2601],{"className":2600},[388],[385,2602,2604,2659],{"className":2603,"ariaHidden":393},[392],[385,2605,2607,2610,2613,2616,2656],{"className":2606},[397],[385,2608],{"className":2609,"style":484},[401],[385,2611,1795],{"className":2612,"style":1794},[406,488],[385,2614],{"className":2615,"style":494},[493],[385,2617,2619,2622],{"className":2618},[498],[385,2620,502],{"className":2621},[498],[385,2623,2625],{"className":2624},[506],[385,2626,2628,2648],{"className":2627},[510,511],[385,2629,2631,2645],{"className":2630},[515],[385,2632,2634],{"className":2633,"style":520},[519],[385,2635,2636,2639],{"style":523},[385,2637],{"className":2638,"style":528},[527],[385,2640,2642],{"className":2641},[532,533,534,535],[385,2643,433],{"className":2644,"style":539},[406,488,535],[385,2646,544],{"className":2647},[543],[385,2649,2651],{"className":2650},[515],[385,2652,2654],{"className":2653,"style":551},[519],[385,2655],{},[385,2657],{"className":2658,"style":494},[493],[385,2660,2662,2665],{"className":2661},[397],[385,2663],{"className":2664,"style":563},[401],[385,2666,1886],{"className":2667,"style":1885},[406,488]," extends the chain, so ",[385,2670,2672],{"className":2671},[388],[385,2673,2675],{"className":2674,"ariaHidden":393},[392],[385,2676,2678,2681],{"className":2677},[397],[385,2679],{"className":2680,"style":563},[401],[385,2682,1886],{"className":2683,"style":1885},[406,488]," inherits hardness from the whole class.",[381,2686,2687,2688,2706,2707,1088,2710,2728],{},"But the engine needs a spark. Transitivity spreads\n",[385,2689,2691],{"className":2690},[388],[385,2692,2694],{"className":2693,"ariaHidden":393},[392],[385,2695,2697,2700],{"className":2696},[397],[385,2698],{"className":2699,"style":402},[401],[385,2701,2703],{"className":2702},[406],[385,2704,411],{"className":2705},[406,410],"-completeness from one problem to the next, yet it presupposes we\nalready have a ",[414,2708,2709],{},"first",[385,2711,2713],{"className":2712},[388],[385,2714,2716],{"className":2715,"ariaHidden":393},[392],[385,2717,2719,2722],{"className":2718},[397],[385,2720],{"className":2721,"style":402},[401],[385,2723,2725],{"className":2724},[406],[385,2726,411],{"className":2727},[406,410],"-complete problem to start from. Where does\nthe first one come from?",[465,2730,2732],{"id":2731},"the-first-one-cooklevin","The first one: Cook–Levin",[381,2734,2735,2736,2754,2755,2758],{},"The breakthrough, proved independently by Stephen Cook (1971) and Leonid Levin,\nis that a natural problem is ",[385,2737,2739],{"className":2738},[388],[385,2740,2742],{"className":2741,"ariaHidden":393},[392],[385,2743,2745,2748],{"className":2744},[397],[385,2746],{"className":2747,"style":402},[401],[385,2749,2751],{"className":2750},[406],[385,2752,411],{"className":2753},[406,410],"-complete ",[414,2756,2757],{},"from scratch",", without\nreducing from anything, by reasoning directly about computation itself.",[381,2760,2761,2762,2765,2766,604],{},"The problem is ",[455,2763,2764],{},"boolean satisfiability",", or ",[455,2767,457],{},[681,2769,2770],{"type":683},[381,2771,2772,2775,2776,2793,2794,2913,2914,2931,2932,2948,2949,2966,2967,2969,2970,2973,2974,2989,2990,2992],{},[455,2773,2774],{},"Definition (SAT)."," Given a boolean formula ",[385,2777,2779],{"className":2778},[388],[385,2780,2782],{"className":2781,"ariaHidden":393},[392],[385,2783,2785,2789],{"className":2784},[397],[385,2786],{"className":2787,"style":2788},[401],"height:0.625em;vertical-align:-0.1944em;",[385,2790,2792],{"className":2791},[406,488],"φ"," over variables ",[385,2795,2797],{"className":2796},[388],[385,2798,2800],{"className":2799,"ariaHidden":393},[392],[385,2801,2803,2806,2848,2853,2857,2862,2865,2868,2871],{"className":2802},[397],[385,2804],{"className":2805,"style":2788},[401],[385,2807,2809,2813],{"className":2808},[406],[385,2810,2812],{"className":2811},[406,488],"x",[385,2814,2816],{"className":2815},[506],[385,2817,2819,2840],{"className":2818},[510,511],[385,2820,2822,2837],{"className":2821},[515],[385,2823,2826],{"className":2824,"style":2825},[519],"height:0.3011em;",[385,2827,2828,2831],{"style":523},[385,2829],{"className":2830,"style":528},[527],[385,2832,2834],{"className":2833},[532,533,534,535],[385,2835,1117],{"className":2836},[406,535],[385,2838,544],{"className":2839},[543],[385,2841,2843],{"className":2842},[515],[385,2844,2846],{"className":2845,"style":551},[519],[385,2847],{},[385,2849,2852],{"className":2850},[2851],"mpunct",",",[385,2854],{"className":2855,"style":2856},[493],"margin-right:0.1667em;",[385,2858,2861],{"className":2859},[2860],"minner","…",[385,2863],{"className":2864,"style":2856},[493],[385,2866,2852],{"className":2867},[2851],[385,2869],{"className":2870,"style":2856},[493],[385,2872,2874,2877],{"className":2873},[406],[385,2875,2812],{"className":2876},[406,488],[385,2878,2880],{"className":2879},[506],[385,2881,2883,2905],{"className":2882},[510,511],[385,2884,2886,2902],{"className":2885},[515],[385,2887,2890],{"className":2888,"style":2889},[519],"height:0.1514em;",[385,2891,2892,2895],{"style":523},[385,2893],{"className":2894,"style":528},[527],[385,2896,2898],{"className":2897},[532,533,534,535],[385,2899,2901],{"className":2900},[406,488,535],"n",[385,2903,544],{"className":2904},[543],[385,2906,2908],{"className":2907},[515],[385,2909,2911],{"className":2910,"style":551},[519],[385,2912],{},"\nusing ",[385,2915,2917],{"className":2916},[388],[385,2918,2920],{"className":2919,"ariaHidden":393},[392],[385,2921,2923,2927],{"className":2922},[397],[385,2924],{"className":2925,"style":2926},[401],"height:0.5556em;",[385,2928,2930],{"className":2929},[406],"∧"," (and), ",[385,2933,2935],{"className":2934},[388],[385,2936,2938],{"className":2937,"ariaHidden":393},[392],[385,2939,2941,2944],{"className":2940},[397],[385,2942],{"className":2943,"style":2926},[401],[385,2945,2947],{"className":2946},[406],"∨"," (or), and ",[385,2950,2952],{"className":2951},[388],[385,2953,2955],{"className":2954,"ariaHidden":393},[392],[385,2956,2958,2962],{"className":2957},[397],[385,2959],{"className":2960,"style":2961},[401],"height:0.4306em;",[385,2963,2965],{"className":2964},[406],"¬"," (not), is there an assignment of\n",[455,2968,393],{},"\u002F",[455,2971,2972],{},"false"," to the variables that makes ",[385,2975,2977],{"className":2976},[388],[385,2978,2980],{"className":2979,"ariaHidden":393},[392],[385,2981,2983,2986],{"className":2982},[397],[385,2984],{"className":2985,"style":2788},[401],[385,2987,2792],{"className":2988},[406,488]," evaluate to ",[455,2991,393],{},"?",[681,2994,2996],{"type":2995},"theorem",[381,2997,2998,1088,3001,743,3003,3021],{},[455,2999,3000],{},"Theorem (Cook–Levin).",[455,3002,457],{},[385,3004,3006],{"className":3005},[388],[385,3007,3009],{"className":3008,"ariaHidden":393},[392],[385,3010,3012,3015],{"className":3011},[397],[385,3013],{"className":3014,"style":402},[401],[385,3016,3018],{"className":3017},[406],[385,3019,411],{"className":3020},[406,410],"-complete.",[381,3023,3024,3025,3027,3028,3046,3047,3062,3063,3081,3082,1088,3084,3120,3121,3123,3124,3127,3128,3164,3165,3182,3183,3198,3199,3233,3234,3256,3257,3309,3310,3362,3363,3366,3367,3382,3383,3385,3386,3438,3439,3454,3455,3470,3471,625,3474,3492,3493,3495,3496,3514,3515],{},"That ",[455,3026,457],{}," is in ",[385,3029,3031],{"className":3030},[388],[385,3032,3034],{"className":3033,"ariaHidden":393},[392],[385,3035,3037,3040],{"className":3036},[397],[385,3038],{"className":3039,"style":402},[401],[385,3041,3043],{"className":3042},[406],[385,3044,411],{"className":3045},[406,410]," is easy: a satisfying assignment is a\ncertificate, and evaluating ",[385,3048,3050],{"className":3049},[388],[385,3051,3053],{"className":3052,"ariaHidden":393},[392],[385,3054,3056,3059],{"className":3055},[397],[385,3057],{"className":3058,"style":2788},[401],[385,3060,2792],{"className":3061},[406,488]," on it takes linear time. The deep half is\n",[385,3064,3066],{"className":3065},[388],[385,3067,3069],{"className":3068,"ariaHidden":393},[392],[385,3070,3072,3075],{"className":3071},[397],[385,3073],{"className":3074,"style":402},[401],[385,3076,3078],{"className":3077},[406],[385,3079,411],{"className":3080},[406,410],"-hardness: showing that ",[414,3083,624],{},[385,3085,3087],{"className":3086},[388],[385,3088,3090,3108],{"className":3089,"ariaHidden":393},[392],[385,3091,3093,3096,3099,3102,3105],{"className":3092},[397],[385,3094],{"className":3095,"style":849},[401],[385,3097,489],{"className":3098},[406,488],[385,3100],{"className":3101,"style":494},[493],[385,3103,859],{"className":3104},[498],[385,3106],{"className":3107,"style":494},[493],[385,3109,3111,3114],{"className":3110},[397],[385,3112],{"className":3113,"style":402},[401],[385,3115,3117],{"className":3116},[406],[385,3118,411],{"className":3119},[406,410]," reduces to\n",[455,3122,457],{},". The idea, which we sketch rather than prove, is to encode ",[414,3125,3126],{},"computation\nas logic",". Any ",[385,3129,3131],{"className":3130},[388],[385,3132,3134,3152],{"className":3133,"ariaHidden":393},[392],[385,3135,3137,3140,3143,3146,3149],{"className":3136},[397],[385,3138],{"className":3139,"style":849},[401],[385,3141,489],{"className":3142},[406,488],[385,3144],{"className":3145,"style":494},[493],[385,3147,859],{"className":3148},[498],[385,3150],{"className":3151,"style":494},[493],[385,3153,3155,3158],{"className":3154},[397],[385,3156],{"className":3157,"style":402},[401],[385,3159,3161],{"className":3160},[406],[385,3162,411],{"className":3163},[406,410]," has a polynomial-time verifier ",[385,3166,3168],{"className":3167},[388],[385,3169,3171],{"className":3170,"ariaHidden":393},[392],[385,3172,3174,3177],{"className":3173},[397],[385,3175],{"className":3176,"style":563},[401],[385,3178,3181],{"className":3179,"style":3180},[406,488],"margin-right:0.2222em;","V",". For an\ninput ",[385,3184,3186],{"className":3185},[388],[385,3187,3189],{"className":3188,"ariaHidden":393},[392],[385,3190,3192,3195],{"className":3191},[397],[385,3193],{"className":3194,"style":2961},[401],[385,3196,2812],{"className":3197},[406,488],", the question ",[571,3200,3201,3202,3217,3218,2992],{},"does some certificate make ",[385,3203,3205],{"className":3204},[388],[385,3206,3208],{"className":3207,"ariaHidden":393},[392],[385,3209,3211,3214],{"className":3210},[397],[385,3212],{"className":3213,"style":563},[401],[385,3215,3181],{"className":3216,"style":3180},[406,488]," accept ",[385,3219,3221],{"className":3220},[388],[385,3222,3224],{"className":3223,"ariaHidden":393},[392],[385,3225,3227,3230],{"className":3226},[397],[385,3228],{"className":3229,"style":2961},[401],[385,3231,2812],{"className":3232},[406,488]," is exactly\nthe question ",[571,3235,3236,3237,3252,3253,2992],{},"is ",[385,3238,3240],{"className":3239},[388],[385,3241,3243],{"className":3242,"ariaHidden":393},[392],[385,3244,3246,3249],{"className":3245},[397],[385,3247],{"className":3248,"style":563},[401],[385,3250,489],{"className":3251},[406,488],"'s answer ",[455,3254,3255],{},"yes"," Cook and Levin build, in polynomial\ntime, a boolean formula ",[385,3258,3260],{"className":3259},[388],[385,3261,3263],{"className":3262,"ariaHidden":393},[392],[385,3264,3266,3269],{"className":3265},[397],[385,3267],{"className":3268,"style":2788},[401],[385,3270,3272,3275],{"className":3271},[406],[385,3273,2792],{"className":3274},[406,488],[385,3276,3278],{"className":3277},[506],[385,3279,3281,3301],{"className":3280},[510,511],[385,3282,3284,3298],{"className":3283},[515],[385,3285,3287],{"className":3286,"style":2889},[519],[385,3288,3289,3292],{"style":523},[385,3290],{"className":3291,"style":528},[527],[385,3293,3295],{"className":3294},[532,533,534,535],[385,3296,2812],{"className":3297},[406,488,535],[385,3299,544],{"className":3300},[543],[385,3302,3304],{"className":3303},[515],[385,3305,3307],{"className":3306,"style":551},[519],[385,3308],{}," whose variables describe the verifier's\nentire step-by-step execution, with clauses that force the variables to obey\nthe machine's rules. Then ",[385,3311,3313],{"className":3312},[388],[385,3314,3316],{"className":3315,"ariaHidden":393},[392],[385,3317,3319,3322],{"className":3318},[397],[385,3320],{"className":3321,"style":2788},[401],[385,3323,3325,3328],{"className":3324},[406],[385,3326,2792],{"className":3327},[406,488],[385,3329,3331],{"className":3330},[506],[385,3332,3334,3354],{"className":3333},[510,511],[385,3335,3337,3351],{"className":3336},[515],[385,3338,3340],{"className":3339,"style":2889},[519],[385,3341,3342,3345],{"style":523},[385,3343],{"className":3344,"style":528},[527],[385,3346,3348],{"className":3347},[532,533,534,535],[385,3349,2812],{"className":3350},[406,488,535],[385,3352,544],{"className":3353},[543],[385,3355,3357],{"className":3356},[515],[385,3358,3360],{"className":3359,"style":551},[519],[385,3361],{}," is satisfiable ",[455,3364,3365],{},"if and only if"," some\ncertificate makes ",[385,3368,3370],{"className":3369},[388],[385,3371,3373],{"className":3372,"ariaHidden":393},[392],[385,3374,3376,3379],{"className":3375},[397],[385,3377],{"className":3378,"style":563},[401],[385,3380,3181],{"className":3381,"style":3180},[406,488]," accept, so deciding ",[455,3384,457],{}," on ",[385,3387,3389],{"className":3388},[388],[385,3390,3392],{"className":3391,"ariaHidden":393},[392],[385,3393,3395,3398],{"className":3394},[397],[385,3396],{"className":3397,"style":2788},[401],[385,3399,3401,3404],{"className":3400},[406],[385,3402,2792],{"className":3403},[406,488],[385,3405,3407],{"className":3406},[506],[385,3408,3410,3430],{"className":3409},[510,511],[385,3411,3413,3427],{"className":3412},[515],[385,3414,3416],{"className":3415,"style":2889},[519],[385,3417,3418,3421],{"style":523},[385,3419],{"className":3420,"style":528},[527],[385,3422,3424],{"className":3423},[532,533,534,535],[385,3425,2812],{"className":3426},[406,488,535],[385,3428,544],{"className":3429},[543],[385,3431,3433],{"className":3432},[515],[385,3434,3436],{"className":3435,"style":551},[519],[385,3437],{}," decides ",[385,3440,3442],{"className":3441},[388],[385,3443,3445],{"className":3444,"ariaHidden":393},[392],[385,3446,3448,3451],{"className":3447},[397],[385,3449],{"className":3450,"style":563},[401],[385,3452,489],{"className":3453},[406,488],"\non ",[385,3456,3458],{"className":3457},[388],[385,3459,3461],{"className":3460,"ariaHidden":393},[392],[385,3462,3464,3467],{"className":3463},[397],[385,3465],{"className":3466,"style":2961},[401],[385,3468,2812],{"className":3469},[406,488],". 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",[385,3937,3939],{"className":3938},[388],[385,3940,3942],{"className":3941,"ariaHidden":393},[392],[385,3943,3945,3948],{"className":3944},[397],[385,3946],{"className":3947,"style":2788},[401],[385,3949,3951,3954],{"className":3950},[406],[385,3952,2792],{"className":3953},[406,488],[385,3955,3957],{"className":3956},[506],[385,3958,3960,3980],{"className":3959},[510,511],[385,3961,3963,3977],{"className":3962},[515],[385,3964,3966],{"className":3965,"style":2889},[519],[385,3967,3968,3971],{"style":523},[385,3969],{"className":3970,"style":528},[527],[385,3972,3974],{"className":3973},[532,533,534,535],[385,3975,2812],{"className":3976},[406,488,535],[385,3978,544],{"className":3979},[543],[385,3981,3983],{"className":3982},[515],[385,3984,3986],{"className":3985,"style":551},[519],[385,3987],{},", satisfiable iff some ",[385,3990,3992],{"className":3991},[388],[385,3993,3995],{"className":3994,"ariaHidden":393},[392],[385,3996,3998,4001],{"className":3997},[397],[385,3999],{"className":4000,"style":2788},[401],[385,4002,3929],{"className":4003,"style":3928},[406,488]," accepts.",[381,4006,4007,4008,4026],{},"With one ",[385,4009,4011],{"className":4010},[388],[385,4012,4014],{"className":4013,"ariaHidden":393},[392],[385,4015,4017,4020],{"className":4016},[397],[385,4018],{"className":4019,"style":402},[401],[385,4021,4023],{"className":4022},[406],[385,4024,411],{"className":4025},[406,410],"-complete problem in hand, the transitivity engine takes\nover.",[465,4028,4030],{"id":4029},"the-reduction-web","The reduction web",[381,4032,4033,4034,4036,4037,4055,4056,4064,4065,4068,4069,4071,4072,4075,4076,4424,4425,1518,4504,4506,4507,4525],{},"Karp's celebrated 1972 paper reduced ",[455,4035,457],{}," to twenty-one other problems,\nshowing them all ",[385,4038,4040],{"className":4039},[388],[385,4041,4043],{"className":4042,"ariaHidden":393},[392],[385,4044,4046,4049],{"className":4045},[397],[385,4047],{"className":4048,"style":402},[401],[385,4050,4052],{"className":4051},[406],[385,4053,411],{"className":4054},[406,410],"-complete and launching the field.",[1109,4057,4058],{},[460,4059,4063],{"href":4060,"ariaDescribedBy":4061,"dataFootnoteRef":376,"id":4062},"#user-content-fn-skiena-web",[1115],"user-content-fnref-skiena-web","3"," A convenient\nwaystation is ",[455,4066,4067],{},"3-SAT",", the special case of ",[455,4070,457],{}," in which the formula is in\n",[414,4073,4074],{},"conjunctive normal form"," with exactly three literals per clause, a big AND of\nsmall ORs such as ",[385,4077,4079],{"className":4078},[388],[385,4080,4082,4141,4200,4258,4319,4374],{"className":4081,"ariaHidden":393},[392],[385,4083,4085,4088,4091,4131,4134,4138],{"className":4084},[397],[385,4086],{"className":4087,"style":3910},[401],[385,4089,3915],{"className":4090},[3914],[385,4092,4094,4097],{"className":4093},[406],[385,4095,2812],{"className":4096},[406,488],[385,4098,4100],{"className":4099},[506],[385,4101,4103,4123],{"className":4102},[510,511],[385,4104,4106,4120],{"className":4105},[515],[385,4107,4109],{"className":4108,"style":2825},[519],[385,4110,4111,4114],{"style":523},[385,4112],{"className":4113,"style":528},[527],[385,4115,4117],{"className":4116},[532,533,534,535],[385,4118,1117],{"className":4119},[406,535],[385,4121,544],{"className":4122},[543],[385,4124,4126],{"className":4125},[515],[385,4127,4129],{"className":4128,"style":551},[519],[385,4130],{},[385,4132],{"className":4133,"style":3180},[493],[385,4135,2947],{"className":4136},[4137],"mbin",[385,4139],{"className":4140,"style":3180},[493],[385,4142,4144,4148,4151,4191,4194,4197],{"className":4143},[397],[385,4145],{"className":4146,"style":4147},[401],"height:0.7056em;vertical-align:-0.15em;",[385,4149,2965],{"className":4150},[406],[385,4152,4154,4157],{"className":4153},[406],[385,4155,2812],{"className":4156},[406,488],[385,4158,4160],{"className":4159},[506],[385,4161,4163,4183],{"className":4162},[510,511],[385,4164,4166,4180],{"className":4165},[515],[385,4167,4169],{"className":4168,"style":2825},[519],[385,4170,4171,4174],{"style":523},[385,4172],{"className":4173,"style":528},[527],[385,4175,4177],{"className":4176},[532,533,534,535],[385,4178,3522],{"className":4179},[406,535],[385,4181,544],{"className":4182},[543],[385,4184,4186],{"className":4185},[515],[385,4187,4189],{"className":4188,"style":551},[519],[385,4190],{},[385,4192],{"className":4193,"style":3180},[493],[385,4195,2947],{"className":4196},[4137],[385,4198],{"className":4199,"style":3180},[493],[385,4201,4203,4206,4246,4249,4252,4255],{"className":4202},[397],[385,4204],{"className":4205,"style":3910},[401],[385,4207,4209,4212],{"className":4208},[406],[385,4210,2812],{"className":4211},[406,488],[385,4213,4215],{"className":4214},[506],[385,4216,4218,4238],{"className":4217},[510,511],[385,4219,4221,4235],{"className":4220},[515],[385,4222,4224],{"className":4223,"style":2825},[519],[385,4225,4226,4229],{"style":523},[385,4227],{"className":4228,"style":528},[527],[385,4230,4232],{"className":4231},[532,533,534,535],[385,4233,4063],{"className":4234},[406,535],[385,4236,544],{"className":4237},[543],[385,4239,4241],{"className":4240},[515],[385,4242,4244],{"className":4243,"style":551},[519],[385,4245],{},[385,4247,3934],{"className":4248},[3933],[385,4250],{"className":4251,"style":3180},[493],[385,4253,2930],{"className":4254},[4137],[385,4256],{"className":4257,"style":3180},[493],[385,4259,4261,4264,4267,4270,4310,4313,4316],{"className":4260},[397],[385,4262],{"className":4263,"style":3910},[401],[385,4265,3915],{"className":4266},[3914],[385,4268,2965],{"className":4269},[406],[385,4271,4273,4276],{"className":4272},[406],[385,4274,2812],{"className":4275},[406,488],[385,4277,4279],{"className":4278},[506],[385,4280,4282,4302],{"className":4281},[510,511],[385,4283,4285,4299],{"className":4284},[515],[385,4286,4288],{"className":4287,"style":2825},[519],[385,4289,4290,4293],{"style":523},[385,4291],{"className":4292,"style":528},[527],[385,4294,4296],{"className":4295},[532,533,534,535],[385,4297,1117],{"className":4298},[406,535],[385,4300,544],{"className":4301},[543],[385,4303,4305],{"className":4304},[515],[385,4306,4308],{"className":4307,"style":551},[519],[385,4309],{},[385,4311],{"className":4312,"style":3180},[493],[385,4314,2947],{"className":4315},[4137],[385,4317],{"className":4318,"style":3180},[493],[385,4320,4322,4325,4365,4368,4371],{"className":4321},[397],[385,4323],{"className":4324,"style":4147},[401],[385,4326,4328,4331],{"className":4327},[406],[385,4329,2812],{"className":4330},[406,488],[385,4332,4334],{"className":4333},[506],[385,4335,4337,4357],{"className":4336},[510,511],[385,4338,4340,4354],{"className":4339},[515],[385,4341,4343],{"className":4342,"style":2825},[519],[385,4344,4345,4348],{"style":523},[385,4346],{"className":4347,"style":528},[527],[385,4349,4351],{"className":4350},[532,533,534,535],[385,4352,3522],{"className":4353},[406,535],[385,4355,544],{"className":4356},[543],[385,4358,4360],{"className":4359},[515],[385,4361,4363],{"className":4362,"style":551},[519],[385,4364],{},[385,4366],{"className":4367,"style":3180},[493],[385,4369,2947],{"className":4370},[4137],[385,4372],{"className":4373,"style":3180},[493],[385,4375,4377,4380,4421],{"className":4376},[397],[385,4378],{"className":4379,"style":3910},[401],[385,4381,4383,4386],{"className":4382},[406],[385,4384,2812],{"className":4385},[406,488],[385,4387,4389],{"className":4388},[506],[385,4390,4392,4413],{"className":4391},[510,511],[385,4393,4395,4410],{"className":4394},[515],[385,4396,4398],{"className":4397,"style":2825},[519],[385,4399,4400,4403],{"style":523},[385,4401],{"className":4402,"style":528},[527],[385,4404,4406],{"className":4405},[532,533,534,535],[385,4407,4409],{"className":4408},[406,535],"4",[385,4411,544],{"className":4412},[543],[385,4414,4416],{"className":4415},[515],[385,4417,4419],{"className":4418,"style":551},[519],[385,4420],{},[385,4422,3934],{"className":4423},[3933],". One can show ",[385,4426,4428],{"className":4427},[388],[385,4429,4431,4491],{"className":4430,"ariaHidden":393},[392],[385,4432,4434,4438,4445,4448,4488],{"className":4433},[397],[385,4435],{"className":4436,"style":4437},[401],"height:0.8361em;vertical-align:-0.15em;",[385,4439,4441],{"className":4440},[406,1567],[385,4442,457],{"className":4443},[406,4444],"textbf",[385,4446],{"className":4447,"style":494},[493],[385,4449,4451,4454],{"className":4450},[498],[385,4452,502],{"className":4453},[498],[385,4455,4457],{"className":4456},[506],[385,4458,4460,4480],{"className":4459},[510,511],[385,4461,4463,4477],{"className":4462},[515],[385,4464,4466],{"className":4465,"style":520},[519],[385,4467,4468,4471],{"style":523},[385,4469],{"className":4470,"style":528},[527],[385,4472,4474],{"className":4473},[532,533,534,535],[385,4475,433],{"className":4476,"style":539},[406,488,535],[385,4478,544],{"className":4479},[543],[385,4481,4483],{"className":4482},[515],[385,4484,4486],{"className":4485,"style":551},[519],[385,4487],{},[385,4489],{"className":4490,"style":494},[493],[385,4492,4494,4498],{"className":4493},[397],[385,4495],{"className":4496,"style":4497},[401],"height:0.6861em;",[385,4499,4501],{"className":4500},[406,1567],[385,4502,4067],{"className":4503},[406,4444],[455,4505,4067],{}," is itself\n",[385,4508,4510],{"className":4509},[388],[385,4511,4513],{"className":4512,"ariaHidden":393},[392],[385,4514,4516,4519],{"className":4515},[397],[385,4517],{"className":4518,"style":402},[401],[385,4520,4522],{"className":4521},[406],[385,4523,411],{"className":4524},[406,410],"-complete, and its rigid structure makes it the favorite\nstarting point for further reductions.",[381,4527,4528,4529,4531],{},"From ",[455,4530,4067],{}," the web branches out. A small portion:",[1119,4533,4535,4702],{"className":4534},[1122,1123],[1125,4536,4540],{"xmlns":1127,"width":4537,"height":4538,"viewBox":4539},"458.558","198.218","-75 -75 343.919 148.664",[1132,4541,4542,4545,4563,4566,4581,4584,4605,4608,4615,4618,4625,4628,4655,4658,4665,4668,4671,4674,4677,4680,4683,4686,4690,4693,4696,4699],{"stroke":1134,"style":1135},[1137,4543],{"fill":1139,"d":4544},"M-42.612 1.907h-22.125a4 4 0 0 0-4 4V26.36a4 4 0 0 0 4 4h22.125a4 4 0 0 0 4-4V5.907a4 4 0 0 0-4-4ZM-68.737 30.36",[1132,4546,4549,4557],{"stroke":1139,"fontFamily":4547,"fontSize":4548},"cmbx10","10",[1132,4550,4552],{"transform":4551},"translate(-11.062 3.43)",[1137,4553],{"d":4554,"fill":1134,"stroke":1134,"className":4555,"style":4556},"M-53.034 16.114L-53.034 13.976Q-53.010 13.878-52.903 13.854L-52.644 13.854Q-52.556 13.878-52.522 13.976Q-52.522 14.615-52.187 15.009Q-51.853 15.402-51.323 15.568Q-50.793 15.734-50.163 15.734Q-49.636 15.734-49.289 15.426Q-48.943 15.118-48.943 14.586Q-48.943 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",[385,4710,4712],{"className":4711},[388],[385,4713,4715,4736],{"className":4714,"ariaHidden":393},[392],[385,4716,4718,4721,4726,4729,4733],{"className":4717},[397],[385,4719],{"className":4720,"style":563},[401],[385,4722,4725],{"className":4723,"style":4724},[406,488],"margin-right:0.0785em;","X",[385,4727],{"className":4728,"style":494},[493],[385,4730,4732],{"className":4731},[498],"→",[385,4734],{"className":4735,"style":494},[493],[385,4737,4739,4742],{"className":4738},[397],[385,4740],{"className":4741,"style":563},[401],[385,4743,4745],{"className":4744,"style":3180},[406,488],"Y"," is a reduction ",[385,4748,4750],{"className":4749},[388],[385,4751,4753,4808],{"className":4752,"ariaHidden":393},[392],[385,4754,4756,4759,4762,4765,4805],{"className":4755},[397],[385,4757],{"className":4758,"style":484},[401],[385,4760,4725],{"className":4761,"style":4724},[406,488],[385,4763],{"className":4764,"style":494},[493],[385,4766,4768,4771],{"className":4767},[498],[385,4769,502],{"className":4770},[498],[385,4772,4774],{"className":4773},[506],[385,4775,4777,4797],{"className":4776},[510,511],[385,4778,4780,4794],{"className":4779},[515],[385,4781,4783],{"className":4782,"style":520},[519],[385,4784,4785,4788],{"style":523},[385,4786],{"className":4787,"style":528},[527],[385,4789,4791],{"className":4790},[532,533,534,535],[385,4792,433],{"className":4793,"style":539},[406,488,535],[385,4795,544],{"className":4796},[543],[385,4798,4800],{"className":4799},[515],[385,4801,4803],{"className":4802,"style":551},[519],[385,4804],{},[385,4806],{"className":4807,"style":494},[493],[385,4809,4811,4814],{"className":4810},[397],[385,4812],{"className":4813,"style":563},[401],[385,4815,4745],{"className":4816,"style":3180},[406,488],"; following arrows back to\n",[455,4819,457],{}," certifies each box as ",[385,4822,4824],{"className":4823},[388],[385,4825,4827],{"className":4826,"ariaHidden":393},[392],[385,4828,4830,4833],{"className":4829},[397],[385,4831],{"className":4832,"style":402},[401],[385,4834,4836],{"className":4835},[406],[385,4837,411],{"className":4838},[406,410],"-complete. (The arrows above record\none ",[414,4841,4842],{},"route"," to each result, not the only one; many of these problems also\nreduce to each other directly, as we saw with ",[385,4845,4847],{"className":4846},[388],[385,4848,4850],{"className":4849,"ariaHidden":393},[392],[385,4851,4853,4856],{"className":4852},[397],[385,4854],{"className":4855,"style":1563},[401],[385,4857,4861],{"className":4858},[4859,4860],"enclosing","textsc",[385,4862,4864],{"className":4863},[406,1567],[385,4865,4867],{"className":4866},[406],"Independent-Set"," and\n",[455,4870,4871],{},"Clique"," last lesson.)",[4874,4875,4877],"h3",{"id":4876},"a-worked-reduction-3-sat-to-independent-set","A worked reduction: 3-SAT to Independent-Set",[381,4879,4880,4881,4957,4958,4973,4974,4990,4991,5007,5008,5025,5026,5041,5042,5045,5046,5061,5062,5077],{},"Let us actually build one arrow, the classic ",[385,4882,4884],{"className":4883},[388],[385,4885,4887,4945],{"className":4886,"ariaHidden":393},[392],[385,4888,4890,4893,4899,4902,4942],{"className":4889},[397],[385,4891],{"className":4892,"style":4437},[401],[385,4894,4896],{"className":4895},[406,1567],[385,4897,4067],{"className":4898},[406,4444],[385,4900],{"className":4901,"style":494},[493],[385,4903,4905,4908],{"className":4904},[498],[385,4906,502],{"className":4907},[498],[385,4909,4911],{"className":4910},[506],[385,4912,4914,4934],{"className":4913},[510,511],[385,4915,4917,4931],{"className":4916},[515],[385,4918,4920],{"className":4919,"style":520},[519],[385,4921,4922,4925],{"style":523},[385,4923],{"className":4924,"style":528},[527],[385,4926,4928],{"className":4927},[532,533,534,535],[385,4929,433],{"className":4930,"style":539},[406,488,535],[385,4932,544],{"className":4933},[543],[385,4935,4937],{"className":4936},[515],[385,4938,4940],{"className":4939,"style":551},[519],[385,4941],{},[385,4943],{"className":4944,"style":494},[493],[385,4946,4948,4951],{"className":4947},[397],[385,4949],{"className":4950,"style":1563},[401],[385,4952,4954],{"className":4953},[406,1567],[385,4955,4867],{"className":4956},[406,4444],". We are given a 3-CNF formula ",[385,4959,4961],{"className":4960},[388],[385,4962,4964],{"className":4963,"ariaHidden":393},[392],[385,4965,4967,4970],{"className":4966},[397],[385,4968],{"className":4969,"style":2788},[401],[385,4971,2792],{"className":4972},[406,488]," with ",[385,4975,4977],{"className":4976},[388],[385,4978,4980],{"className":4979,"ariaHidden":393},[392],[385,4981,4983,4986],{"className":4982},[397],[385,4984],{"className":4985,"style":2961},[401],[385,4987,4989],{"className":4988},[406,488],"m","\nclauses and must produce a graph ",[385,4992,4994],{"className":4993},[388],[385,4995,4997],{"className":4996,"ariaHidden":393},[392],[385,4998,5000,5003],{"className":4999},[397],[385,5001],{"className":5002,"style":563},[401],[385,5004,5006],{"className":5005},[406,488],"G"," and integer ",[385,5009,5011],{"className":5010},[388],[385,5012,5014],{"className":5013,"ariaHidden":393},[392],[385,5015,5017,5020],{"className":5016},[397],[385,5018],{"className":5019,"style":402},[401],[385,5021,5024],{"className":5022,"style":5023},[406,488],"margin-right:0.0315em;","k"," such that ",[385,5027,5029],{"className":5028},[388],[385,5030,5032],{"className":5031,"ariaHidden":393},[392],[385,5033,5035,5038],{"className":5034},[397],[385,5036],{"className":5037,"style":563},[401],[385,5039,5006],{"className":5040},[406,488]," has an\n",[460,5043,5044],{"href":158},"independent set"," of size ",[385,5047,5049],{"className":5048},[388],[385,5050,5052],{"className":5051,"ariaHidden":393},[392],[385,5053,5055,5058],{"className":5054},[397],[385,5056],{"className":5057,"style":402},[401],[385,5059,5024],{"className":5060,"style":5023},[406,488]," exactly when ",[385,5063,5065],{"className":5064},[388],[385,5066,5068],{"className":5067,"ariaHidden":393},[392],[385,5069,5071,5074],{"className":5070},[397],[385,5072],{"className":5073,"style":2788},[401],[385,5075,2792],{"className":5076},[406,488]," is satisfiable.",[381,5079,5080,5083,5084,5087,5088,5259,5260,3492,5313,3492,5368,5420,5421,5424,5425,5428,5429,5483,5484,5539,5540,5573,5574,5593,5594,5648],{},[455,5081,5082],{},"The construction."," For each clause, create a ",[414,5085,5086],{},"triangle"," of three vertices,\none per literal. So clause ",[385,5089,5091],{"className":5090},[388],[385,5092,5094,5152,5210],{"className":5093,"ariaHidden":393},[392],[385,5095,5097,5100,5103,5143,5146,5149],{"className":5096},[397],[385,5098],{"className":5099,"style":3910},[401],[385,5101,3915],{"className":5102},[3914],[385,5104,5106,5109],{"className":5105},[406],[385,5107,2812],{"className":5108},[406,488],[385,5110,5112],{"className":5111},[506],[385,5113,5115,5135],{"className":5114},[510,511],[385,5116,5118,5132],{"className":5117},[515],[385,5119,5121],{"className":5120,"style":2825},[519],[385,5122,5123,5126],{"style":523},[385,5124],{"className":5125,"style":528},[527],[385,5127,5129],{"className":5128},[532,533,534,535],[385,5130,1117],{"className":5131},[406,535],[385,5133,544],{"className":5134},[543],[385,5136,5138],{"className":5137},[515],[385,5139,5141],{"className":5140,"style":551},[519],[385,5142],{},[385,5144],{"className":5145,"style":3180},[493],[385,5147,2947],{"className":5148},[4137],[385,5150],{"className":5151,"style":3180},[493],[385,5153,5155,5158,5161,5201,5204,5207],{"className":5154},[397],[385,5156],{"className":5157,"style":4147},[401],[385,5159,2965],{"className":5160},[406],[385,5162,5164,5167],{"className":5163},[406],[385,5165,2812],{"className":5166},[406,488],[385,5168,5170],{"className":5169},[506],[385,5171,5173,5193],{"className":5172},[510,511],[385,5174,5176,5190],{"className":5175},[515],[385,5177,5179],{"className":5178,"style":2825},[519],[385,5180,5181,5184],{"style":523},[385,5182],{"className":5183,"style":528},[527],[385,5185,5187],{"className":5186},[532,533,534,535],[385,5188,3522],{"className":5189},[406,535],[385,5191,544],{"className":5192},[543],[385,5194,5196],{"className":5195},[515],[385,5197,5199],{"className":5198,"style":551},[519],[385,5200],{},[385,5202],{"className":5203,"style":3180},[493],[385,5205,2947],{"className":5206},[4137],[385,5208],{"className":5209,"style":3180},[493],[385,5211,5213,5216,5256],{"className":5212},[397],[385,5214],{"className":5215,"style":3910},[401],[385,5217,5219,5222],{"className":5218},[406],[385,5220,2812],{"className":5221},[406,488],[385,5223,5225],{"className":5224},[506],[385,5226,5228,5248],{"className":5227},[510,511],[385,5229,5231,5245],{"className":5230},[515],[385,5232,5234],{"className":5233,"style":2825},[519],[385,5235,5236,5239],{"style":523},[385,5237],{"className":5238,"style":528},[527],[385,5240,5242],{"className":5241},[532,533,534,535],[385,5243,4063],{"className":5244},[406,535],[385,5246,544],{"className":5247},[543],[385,5249,5251],{"className":5250},[515],[385,5252,5254],{"className":5253,"style":551},[519],[385,5255],{},[385,5257,3934],{"className":5258},[3933]," becomes three\nmutually-connected vertices labeled ",[385,5261,5263],{"className":5262},[388],[385,5264,5266],{"className":5265,"ariaHidden":393},[392],[385,5267,5269,5273],{"className":5268},[397],[385,5270],{"className":5271,"style":5272},[401],"height:0.5806em;vertical-align:-0.15em;",[385,5274,5276,5279],{"className":5275},[406],[385,5277,2812],{"className":5278},[406,488],[385,5280,5282],{"className":5281},[506],[385,5283,5285,5305],{"className":5284},[510,511],[385,5286,5288,5302],{"className":5287},[515],[385,5289,5291],{"className":5290,"style":2825},[519],[385,5292,5293,5296],{"style":523},[385,5294],{"className":5295,"style":528},[527],[385,5297,5299],{"className":5298},[532,533,534,535],[385,5300,1117],{"className":5301},[406,535],[385,5303,544],{"className":5304},[543],[385,5306,5308],{"className":5307},[515],[385,5309,5311],{"className":5310,"style":551},[519],[385,5312],{},[385,5314,5316],{"className":5315},[388],[385,5317,5319],{"className":5318,"ariaHidden":393},[392],[385,5320,5322,5325,5328],{"className":5321},[397],[385,5323],{"className":5324,"style":5272},[401],[385,5326,2965],{"className":5327},[406],[385,5329,5331,5334],{"className":5330},[406],[385,5332,2812],{"className":5333},[406,488],[385,5335,5337],{"className":5336},[506],[385,5338,5340,5360],{"className":5339},[510,511],[385,5341,5343,5357],{"className":5342},[515],[385,5344,5346],{"className":5345,"style":2825},[519],[385,5347,5348,5351],{"style":523},[385,5349],{"className":5350,"style":528},[527],[385,5352,5354],{"className":5353},[532,533,534,535],[385,5355,3522],{"className":5356},[406,535],[385,5358,544],{"className":5359},[543],[385,5361,5363],{"className":5362},[515],[385,5364,5366],{"className":5365,"style":551},[519],[385,5367],{},[385,5369,5371],{"className":5370},[388],[385,5372,5374],{"className":5373,"ariaHidden":393},[392],[385,5375,5377,5380],{"className":5376},[397],[385,5378],{"className":5379,"style":5272},[401],[385,5381,5383,5386],{"className":5382},[406],[385,5384,2812],{"className":5385},[406,488],[385,5387,5389],{"className":5388},[506],[385,5390,5392,5412],{"className":5391},[510,511],[385,5393,5395,5409],{"className":5394},[515],[385,5396,5398],{"className":5397,"style":2825},[519],[385,5399,5400,5403],{"style":523},[385,5401],{"className":5402,"style":528},[527],[385,5404,5406],{"className":5405},[532,533,534,535],[385,5407,4063],{"className":5408},[406,535],[385,5410,544],{"className":5411},[543],[385,5413,5415],{"className":5414},[515],[385,5416,5418],{"className":5417,"style":551},[519],[385,5419],{},". Then add a\n",[455,5422,5423],{},"conflict edge"," between any two vertices in different triangles that hold\n",[414,5426,5427],{},"contradictory"," literals: one labeled ",[385,5430,5432],{"className":5431},[388],[385,5433,5435],{"className":5434,"ariaHidden":393},[392],[385,5436,5438,5441],{"className":5437},[397],[385,5439],{"className":5440,"style":5272},[401],[385,5442,5444,5447],{"className":5443},[406],[385,5445,2812],{"className":5446},[406,488],[385,5448,5450],{"className":5449},[506],[385,5451,5453,5475],{"className":5452},[510,511],[385,5454,5456,5472],{"className":5455},[515],[385,5457,5460],{"className":5458,"style":5459},[519],"height:0.3117em;",[385,5461,5462,5465],{"style":523},[385,5463],{"className":5464,"style":528},[527],[385,5466,5468],{"className":5467},[532,533,534,535],[385,5469,5471],{"className":5470},[406,488,535],"i",[385,5473,544],{"className":5474},[543],[385,5476,5478],{"className":5477},[515],[385,5479,5481],{"className":5480,"style":551},[519],[385,5482],{}," and another labeled ",[385,5485,5487],{"className":5486},[388],[385,5488,5490],{"className":5489,"ariaHidden":393},[392],[385,5491,5493,5496,5499],{"className":5492},[397],[385,5494],{"className":5495,"style":5272},[401],[385,5497,2965],{"className":5498},[406],[385,5500,5502,5505],{"className":5501},[406],[385,5503,2812],{"className":5504},[406,488],[385,5506,5508],{"className":5507},[506],[385,5509,5511,5531],{"className":5510},[510,511],[385,5512,5514,5528],{"className":5513},[515],[385,5515,5517],{"className":5516,"style":5459},[519],[385,5518,5519,5522],{"style":523},[385,5520],{"className":5521,"style":528},[527],[385,5523,5525],{"className":5524},[532,533,534,535],[385,5526,5471],{"className":5527},[406,488,535],[385,5529,544],{"className":5530},[543],[385,5532,5534],{"className":5533},[515],[385,5535,5537],{"className":5536,"style":551},[519],[385,5538],{},".\nFinally set ",[385,5541,5543],{"className":5542},[388],[385,5544,5546,5564],{"className":5545,"ariaHidden":393},[392],[385,5547,5549,5552,5555,5558,5561],{"className":5548},[397],[385,5550],{"className":5551,"style":402},[401],[385,5553,5024],{"className":5554,"style":5023},[406,488],[385,5556],{"className":5557,"style":494},[493],[385,5559,1635],{"className":5560},[498],[385,5562],{"className":5563,"style":494},[493],[385,5565,5567,5570],{"className":5566},[397],[385,5568],{"className":5569,"style":2961},[401],[385,5571,4989],{"className":5572},[406,488],", the number of clauses. This is clearly polynomial: ",[385,5575,5577],{"className":5576},[388],[385,5578,5580],{"className":5579,"ariaHidden":393},[392],[385,5581,5583,5587,5590],{"className":5582},[397],[385,5584],{"className":5585,"style":5586},[401],"height:0.6444em;",[385,5588,4063],{"className":5589},[406],[385,5591,4989],{"className":5592},[406,488],"\nvertices and at most ",[385,5595,5597],{"className":5596},[388],[385,5598,5600],{"className":5599,"ariaHidden":393},[392],[385,5601,5603,5607,5611,5614,5645],{"className":5602},[397],[385,5604],{"className":5605,"style":5606},[401],"height:1.0641em;vertical-align:-0.25em;",[385,5608,5610],{"className":5609,"style":1885},[406,488],"O",[385,5612,3915],{"className":5613},[3914],[385,5615,5617,5620],{"className":5616},[406],[385,5618,4989],{"className":5619},[406,488],[385,5621,5623],{"className":5622},[506],[385,5624,5626],{"className":5625},[510],[385,5627,5629],{"className":5628},[515],[385,5630,5633],{"className":5631,"style":5632},[519],"height:0.8141em;",[385,5634,5636,5639],{"style":5635},"top:-3.063em;margin-right:0.05em;",[385,5637],{"className":5638,"style":528},[527],[385,5640,5642],{"className":5641},[532,533,534,535],[385,5643,3522],{"className":5644},[406,535],[385,5646,3934],{"className":5647},[3933]," edges.",[381,5650,5651,5654,5655,5658,5659,5692,5693,5696,5697,1268,5749,5804,5805,5807,5808,5823,5824,5839,5840,5900,5901],{},[455,5652,5653],{},"Why it works."," An independent set may pick ",[414,5656,5657],{},"at most one"," vertex from each\ntriangle (the three are mutually adjacent), so an independent set of size ",[385,5660,5662],{"className":5661},[388],[385,5663,5665,5683],{"className":5664,"ariaHidden":393},[392],[385,5666,5668,5671,5674,5677,5680],{"className":5667},[397],[385,5669],{"className":5670,"style":402},[401],[385,5672,5024],{"className":5673,"style":5023},[406,488],[385,5675],{"className":5676,"style":494},[493],[385,5678,1635],{"className":5679},[498],[385,5681],{"className":5682,"style":494},[493],[385,5684,5686,5689],{"className":5685},[397],[385,5687],{"className":5688,"style":2961},[401],[385,5690,4989],{"className":5691},[406,488]," must pick ",[414,5694,5695],{},"exactly one"," literal from every clause. The conflict edges forbid\nchoosing both ",[385,5698,5700],{"className":5699},[388],[385,5701,5703],{"className":5702,"ariaHidden":393},[392],[385,5704,5706,5709],{"className":5705},[397],[385,5707],{"className":5708,"style":5272},[401],[385,5710,5712,5715],{"className":5711},[406],[385,5713,2812],{"className":5714},[406,488],[385,5716,5718],{"className":5717},[506],[385,5719,5721,5741],{"className":5720},[510,511],[385,5722,5724,5738],{"className":5723},[515],[385,5725,5727],{"className":5726,"style":5459},[519],[385,5728,5729,5732],{"style":523},[385,5730],{"className":5731,"style":528},[527],[385,5733,5735],{"className":5734},[532,533,534,535],[385,5736,5471],{"className":5737},[406,488,535],[385,5739,544],{"className":5740},[543],[385,5742,5744],{"className":5743},[515],[385,5745,5747],{"className":5746,"style":551},[519],[385,5748],{},[385,5750,5752],{"className":5751},[388],[385,5753,5755],{"className":5754,"ariaHidden":393},[392],[385,5756,5758,5761,5764],{"className":5757},[397],[385,5759],{"className":5760,"style":5272},[401],[385,5762,2965],{"className":5763},[406],[385,5765,5767,5770],{"className":5766},[406],[385,5768,2812],{"className":5769},[406,488],[385,5771,5773],{"className":5772},[506],[385,5774,5776,5796],{"className":5775},[510,511],[385,5777,5779,5793],{"className":5778},[515],[385,5780,5782],{"className":5781,"style":5459},[519],[385,5783,5784,5787],{"style":523},[385,5785],{"className":5786,"style":528},[527],[385,5788,5790],{"className":5789},[532,533,534,535],[385,5791,5471],{"className":5792},[406,488,535],[385,5794,544],{"className":5795},[543],[385,5797,5799],{"className":5798},[515],[385,5800,5802],{"className":5801,"style":551},[519],[385,5803],{}," anywhere, so the chosen literals are\nmutually consistent — they describe a partial truth assignment. Setting each\nchosen literal ",[455,5806,393],{}," satisfies one literal in every clause, hence satisfies\n",[385,5809,5811],{"className":5810},[388],[385,5812,5814],{"className":5813,"ariaHidden":393},[392],[385,5815,5817,5820],{"className":5816},[397],[385,5818],{"className":5819,"style":2788},[401],[385,5821,2792],{"className":5822},[406,488],". Run the argument backward: a satisfying assignment picks, from each\nclause, one true literal; those ",[385,5825,5827],{"className":5826},[388],[385,5828,5830],{"className":5829,"ariaHidden":393},[392],[385,5831,5833,5836],{"className":5832},[397],[385,5834],{"className":5835,"style":2961},[401],[385,5837,4989],{"className":5838},[406,488]," vertices form an independent set, since two\ntrue literals can never be a variable and its negation. Therefore\n",[385,5841,5843],{"className":5842},[388],[385,5844,5846,5878],{"className":5845,"ariaHidden":393},[392],[385,5847,5849,5852,5855,5862,5865,5868,5872,5875],{"className":5848},[397],[385,5850],{"className":5851,"style":1563},[401],[385,5853,2792],{"className":5854},[406,488],[385,5856,5858],{"className":5857},[406,1567],[385,5859,5861],{"className":5860},[406]," is satisfiable",[385,5863],{"className":5864,"style":494},[493],[385,5866],{"className":5867,"style":494},[493],[385,5869,5871],{"className":5870},[498],"↔",[385,5873],{"className":5874,"style":494},[493],[385,5876],{"className":5877,"style":494},[493],[385,5879,5881,5884,5887,5894,5897],{"className":5880},[397],[385,5882],{"className":5883,"style":1563},[401],[385,5885,5006],{"className":5886},[406,488],[385,5888,5890],{"className":5889},[406,1567],[385,5891,5893],{"className":5892},[406]," has an independent set of size ",[385,5895,4989],{"className":5896},[406,488],[385,5898,2852],{"className":5899},[2851],"\nwhich is exactly what a valid reduction requires.",[1109,5902,5903],{},[460,5904,4409],{"href":5905,"ariaDescribedBy":5906,"dataFootnoteRef":376,"id":5907},"#user-content-fn-erickson-gadget",[1115],"user-content-fnref-erickson-gadget",[1119,5909,5911,6047],{"className":5910},[1122,1123],[1125,5912,5916],{"xmlns":1127,"width":5913,"height":5914,"viewBox":5915},"242.652","95.119","-75 -75 181.989 71.339",[1132,5917,5918,5921,5936,5939,5960,5963,5977,5980,5999,6002,6016,6019,6033,6036,6040],{"stroke":1134,"style":1135},[1137,5919],{"fill":1139,"d":5920},"M-30.209-55.853a8.536 8.536 0 1 0-17.071 0 8.536 8.536 0 0 0 17.071 0Zm-8.536 0",[1132,5922,5923,5930],{"stroke":1139},[1132,5924,5926],{"transform":5925},"translate(-4.705 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206.290-22.771Q206.383-22.624 206.383-22.453Q206.383-22.258 206.280-22.073Q206.178-21.889 206.002-21.773Q205.826-21.656 205.627-21.656Q205.497-21.656 205.308-21.706Q205.118-21.756 205.118-21.851M208.266-22.337Q207.941-22.337 207.697-22.494Q207.453-22.651 207.321-22.916Q207.189-23.181 207.189-23.506Q207.189-23.974 207.442-24.437Q207.695-24.900 208.119-25.196Q208.543-25.491 209.015-25.491Q209.230-25.491 209.416-25.387Q209.602-25.283 209.722-25.098Q209.746-25.211 209.840-25.285Q209.934-25.358 210.050-25.358Q210.153-25.358 210.221-25.297Q210.289-25.235 210.289-25.136Q210.289-25.078 210.283-25.051L209.801-23.133Q209.773-22.976 209.773-22.887Q209.773-22.760 209.825-22.660Q209.876-22.559 209.995-22.559Q210.218-22.559 210.330-22.812Q210.443-23.065 210.529-23.434Q210.553-23.495 210.604-23.495L210.717-23.495Q210.751-23.495 210.773-23.466Q210.795-23.437 210.795-23.413Q210.795-23.400 210.788-23.386Q210.525-22.337 209.982-22.337Q209.732-22.337 209.524-22.460Q209.315-22.583 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212.697-25.365Q212.911-25.238 212.973-25.016Q213.181-25.232 213.434-25.362Q213.687-25.491 213.950-25.491Q214.271-25.491 214.517-25.336Q214.764-25.180 214.893-24.914Q215.023-24.647 215.023-24.322Q215.023-23.970 214.878-23.618Q214.733-23.266 214.482-22.978Q214.230-22.689 213.895-22.513Q213.560-22.337 213.202-22.337Q212.750-22.337 212.497-22.726L212.193-21.523Q212.173-21.462 212.173-21.397Q212.173-21.328 212.614-21.328Q212.699-21.301 212.699-21.216L212.672-21.103Q212.644-21.055 212.593-21.048M212.973-24.630L212.600-23.154Q212.661-22.904 212.819-22.731Q212.976-22.559 213.215-22.559Q213.424-22.559 213.613-22.685Q213.803-22.812 213.940-23.001Q214.077-23.191 214.162-23.393Q214.268-23.656 214.352-24.023Q214.435-24.391 214.435-24.623Q214.435-24.777 214.384-24.929Q214.333-25.081 214.220-25.175Q214.107-25.269 213.936-25.269Q213.656-25.269 213.412-25.086Q213.167-24.904 212.973-24.630M215.645-23.560Q215.645-24.056 215.943-24.505Q216.240-24.955 216.712-25.223Q217.183-25.491 217.676-25.491Q217.942-25.491 218.173-25.408Q218.404-25.324 218.581-25.160Q218.759-24.996 218.858-24.769Q218.957-24.541 218.957-24.268Q218.957-23.772 218.660-23.323Q218.363-22.873 217.893-22.605Q217.423-22.337 216.934-22.337Q216.578-22.337 216.281-22.490Q215.984-22.644 215.815-22.923Q215.645-23.201 215.645-23.560M216.948-22.559Q217.272-22.559 217.539-22.743Q217.806-22.928 217.987-23.227Q218.168-23.526 218.260-23.873Q218.352-24.220 218.352-24.517Q218.352-24.849 218.170-25.059Q217.987-25.269 217.662-25.269Q217.330-25.269 217.066-25.085Q216.801-24.900 216.621-24.599Q216.442-24.299 216.348-23.953Q216.254-23.608 216.254-23.307Q216.254-22.986 216.442-22.772Q216.630-22.559 216.948-22.559M220-22.825Q220.191-22.559 220.796-22.559Q221.025-22.559 221.270-22.629Q221.514-22.699 221.676-22.854Q221.839-23.010 221.839-23.253Q221.839-23.427 221.688-23.535Q221.538-23.642 221.343-23.680L220.889-23.762Q220.619-23.817 220.431-24.005Q220.243-24.193 220.243-24.449Q220.243-24.774 220.429-25.011Q220.615-25.249 220.912-25.370Q221.210-25.491 221.531-25.491Q221.736-25.491 221.952-25.432Q222.167-25.372 222.309-25.237Q222.451-25.102 222.451-24.890Q222.451-24.722 222.357-24.596Q222.263-24.469 222.099-24.469Q221.999-24.469 221.926-24.534Q221.852-24.599 221.852-24.702Q221.852-24.822 221.936-24.919Q222.020-25.016 222.133-25.037Q221.975-25.269 221.517-25.269Q221.220-25.269 220.971-25.126Q220.721-24.982 220.721-24.702Q220.721-24.446 221.083-24.363L221.545-24.281Q221.859-24.220 222.088-24.011Q222.317-23.803 222.317-23.495Q222.317-23.232 222.162-22.981Q222.006-22.730 221.770-22.579Q221.377-22.337 220.789-22.337Q220.362-22.337 220.012-22.492Q219.661-22.648 219.661-23.013Q219.661-23.208 219.778-23.352Q219.894-23.495 220.089-23.495Q220.208-23.495 220.289-23.424Q220.369-23.352 220.369-23.232Q220.369-23.075 220.263-22.959Q220.157-22.843 220-22.825M223.807-21.175Q223.807-21.209 223.835-21.236Q224.101-21.458 224.252-21.785Q224.402-22.111 224.402-22.467L224.402-22.504Q224.300-22.405 224.129-22.405Q223.951-22.405 223.830-22.526Q223.708-22.648 223.708-22.825Q223.708-22.996 223.830-23.118Q223.951-23.239 224.129-23.239Q224.389-23.239 224.508-23Q224.628-22.760 224.628-22.467Q224.628-22.063 224.457-21.694Q224.286-21.325 223.989-21.069Q223.958-21.048 223.934-21.048Q223.890-21.048 223.849-21.089Q223.807-21.130 223.807-21.175",[1149],[1227,6554,6556,6557,6663,6664,6679],{"className":6555},[1230],"Choosing ",[385,6558,6560],{"className":6559},[388],[385,6561,6563],{"className":6562,"ariaHidden":393},[392],[385,6564,6566,6569,6573,6613,6616,6619,6659],{"className":6565},[397],[385,6567],{"className":6568,"style":3910},[401],[385,6570,6572],{"className":6571},[3914],"{",[385,6574,6576,6579],{"className":6575},[406],[385,6577,2812],{"className":6578},[406,488],[385,6580,6582],{"className":6581},[506],[385,6583,6585,6605],{"className":6584},[510,511],[385,6586,6588,6602],{"className":6587},[515],[385,6589,6591],{"className":6590,"style":2825},[519],[385,6592,6593,6596],{"style":523},[385,6594],{"className":6595,"style":528},[527],[385,6597,6599],{"className":6598},[532,533,534,535],[385,6600,1117],{"className":6601},[406,535],[385,6603,544],{"className":6604},[543],[385,6606,6608],{"className":6607},[515],[385,6609,6611],{"className":6610,"style":551},[519],[385,6612],{},[385,6614,2852],{"className":6615},[2851],[385,6617],{"className":6618,"style":2856},[493],[385,6620,6622,6625],{"className":6621},[406],[385,6623,2812],{"className":6624},[406,488],[385,6626,6628],{"className":6627},[506],[385,6629,6631,6651],{"className":6630},[510,511],[385,6632,6634,6648],{"className":6633},[515],[385,6635,6637],{"className":6636,"style":2825},[519],[385,6638,6639,6642],{"style":523},[385,6640],{"className":6641,"style":528},[527],[385,6643,6645],{"className":6644},[532,533,534,535],[385,6646,3522],{"className":6647},[406,535],[385,6649,544],{"className":6650},[543],[385,6652,6654],{"className":6653},[515],[385,6655,6657],{"className":6656,"style":551},[519],[385,6658],{},[385,6660,6662],{"className":6661},[3933],"}"," — one vertex per triangle, no conflict edge between them — is a size-",[385,6665,6667],{"className":6666},[388],[385,6668,6670],{"className":6669,"ariaHidden":393},[392],[385,6671,6673,6676],{"className":6672},[397],[385,6674],{"className":6675,"style":5586},[401],[385,6677,3522],{"className":6678},[406]," independent set, i.e. a satisfying assignment.",[465,6681,6683],{"id":6682},"the-recipe-proving-a-new-problem-np-complete","The recipe: proving a new problem NP-complete",[381,6685,6686,6687,6705,6706,836,6709,6724,6725,6761],{},"Once a stockpile of ",[385,6688,6690],{"className":6689},[388],[385,6691,6693],{"className":6692,"ariaHidden":393},[392],[385,6694,6696,6699],{"className":6695},[397],[385,6697],{"className":6698,"style":402},[401],[385,6700,6702],{"className":6701},[406],[385,6703,411],{"className":6704},[406,410],"-complete problems exists, classifying a\n",[414,6707,6708],{},"new",[385,6710,6712],{"className":6711},[388],[385,6713,6715],{"className":6714,"ariaHidden":393},[392],[385,6716,6718,6721],{"className":6717},[397],[385,6719],{"className":6720,"style":563},[401],[385,6722,4725],{"className":6723,"style":4724},[406,488]," follows a fixed procedure. To prove ",[455,6726,6727,1372,6742,6760],{},[385,6728,6730],{"className":6729},[388],[385,6731,6733],{"className":6732,"ariaHidden":393},[392],[385,6734,6736,6739],{"className":6735},[397],[385,6737],{"className":6738,"style":563},[401],[385,6740,4725],{"className":6741,"style":4724},[406,488],[385,6743,6745],{"className":6744},[388],[385,6746,6748],{"className":6747,"ariaHidden":393},[392],[385,6749,6751,6754],{"className":6750},[397],[385,6752],{"className":6753,"style":402},[401],[385,6755,6757],{"className":6756},[406],[385,6758,411],{"className":6759},[406,410],"-complete",", carry out four steps.",[6763,6764,6765,6855,7031,7294],"ol",{},[1332,6766,6767,6806,6807,6809,6810,6828,6829,6832,6833,6851,6852,604],{},[455,6768,6769,6770,604],{},"Show ",[385,6771,6773],{"className":6772},[388],[385,6774,6776,6794],{"className":6775,"ariaHidden":393},[392],[385,6777,6779,6782,6785,6788,6791],{"className":6778},[397],[385,6780],{"className":6781,"style":849},[401],[385,6783,4725],{"className":6784,"style":4724},[406,488],[385,6786],{"className":6787,"style":494},[493],[385,6789,859],{"className":6790},[498],[385,6792],{"className":6793,"style":494},[493],[385,6795,6797,6800],{"className":6796},[397],[385,6798],{"className":6799,"style":402},[401],[385,6801,6803],{"className":6802},[406],[385,6804,411],{"className":6805},[406,410]," Describe a polynomial-size certificate for\n",[455,6808,3255],{},"-instances and argue it can be checked in polynomial time. This is\nusually the easy step, but skipping it is a real error: an\n",[385,6811,6813],{"className":6812},[388],[385,6814,6816],{"className":6815,"ariaHidden":393},[392],[385,6817,6819,6822],{"className":6818},[397],[385,6820],{"className":6821,"style":402},[401],[385,6823,6825],{"className":6824},[406],[385,6826,411],{"className":6827},[406,410],"-",[414,6830,6831],{},"hard"," problem outside ",[385,6834,6836],{"className":6835},[388],[385,6837,6839],{"className":6838,"ariaHidden":393},[392],[385,6840,6842,6845],{"className":6841},[397],[385,6843],{"className":6844,"style":402},[401],[385,6846,6848],{"className":6847},[406],[385,6849,411],{"className":6850},[406,410]," is hard but not\n",[414,6853,6854],{},"complete",[1332,6856,6857,6894,6895,6898,6899,6914,6915,6917,6918,6940,6941,6962,6963,6940,6985,7007,7008,7030],{},[455,6858,6859,6860,6878,6879],{},"Choose a known ",[385,6861,6863],{"className":6862},[388],[385,6864,6866],{"className":6865,"ariaHidden":393},[392],[385,6867,6869,6872],{"className":6868},[397],[385,6870],{"className":6871,"style":402},[401],[385,6873,6875],{"className":6874},[406],[385,6876,411],{"className":6877},[406,410],"-complete problem ",[385,6880,6882],{"className":6881},[388],[385,6883,6885],{"className":6884,"ariaHidden":393},[392],[385,6886,6888,6891],{"className":6887},[397],[385,6889],{"className":6890,"style":563},[401],[385,6892,4745],{"className":6893,"style":3180},[406,488]," to reduce ",[414,6896,6897],{},"from",".\nPick one whose structure resembles ",[385,6900,6902],{"className":6901},[388],[385,6903,6905],{"className":6904,"ariaHidden":393},[392],[385,6906,6908,6911],{"className":6907},[397],[385,6909],{"className":6910,"style":563},[401],[385,6912,4725],{"className":6913,"style":4724},[406,488]," — ",[455,6916,4067],{}," for logical or\ngadget-style constraints, ",[385,6919,6921],{"className":6920},[388],[385,6922,6924],{"className":6923,"ariaHidden":393},[392],[385,6925,6927,6930],{"className":6926},[397],[385,6928],{"className":6929,"style":563},[401],[385,6931,6933],{"className":6932},[4859,4860],[385,6934,6936],{"className":6935},[406,1567],[385,6937,6939],{"className":6938},[406],"Vertex-Cover"," or ",[385,6942,6944],{"className":6943},[388],[385,6945,6947],{"className":6946,"ariaHidden":393},[392],[385,6948,6950,6953],{"className":6949},[397],[385,6951],{"className":6952,"style":1563},[401],[385,6954,6956],{"className":6955},[4859,4860],[385,6957,6959],{"className":6958},[406,1567],[385,6960,4867],{"className":6961},[406]," for graph\nselection, ",[385,6964,6966],{"className":6965},[388],[385,6967,6969],{"className":6968,"ariaHidden":393},[392],[385,6970,6972,6975],{"className":6971},[397],[385,6973],{"className":6974,"style":402},[401],[385,6976,6978],{"className":6977},[4859,4860],[385,6979,6981],{"className":6980},[406,1567],[385,6982,6984],{"className":6983},[406],"Subset-Sum",[385,6986,6988],{"className":6987},[388],[385,6989,6991],{"className":6990,"ariaHidden":393},[392],[385,6992,6994,6997],{"className":6993},[397],[385,6995],{"className":6996,"style":563},[401],[385,6998,7000],{"className":6999},[4859,4860],[385,7001,7003],{"className":7002},[406,1567],[385,7004,7006],{"className":7005},[406],"Partition"," for numeric targets,\n",[385,7009,7011],{"className":7010},[388],[385,7012,7014],{"className":7013,"ariaHidden":393},[392],[385,7015,7017,7020],{"className":7016},[397],[385,7018],{"className":7019,"style":1563},[401],[385,7021,7023],{"className":7022},[4859,4860],[385,7024,7026],{"className":7025},[406,1567],[385,7027,7029],{"className":7028},[406],"Hamiltonian-Cycle"," for routing.",[1332,7032,7033,7106,7107,7122,7123,7138,7139,7142,7143,7158,7159,7174,7175,7190,7191,7206,7207,7277,7278,7293],{},[455,7034,7035,7036,604],{},"Give a polynomial-time reduction ",[385,7037,7039],{"className":7038},[388],[385,7040,7042,7097],{"className":7041,"ariaHidden":393},[392],[385,7043,7045,7048,7051,7054,7094],{"className":7044},[397],[385,7046],{"className":7047,"style":484},[401],[385,7049,4745],{"className":7050,"style":3180},[406,488],[385,7052],{"className":7053,"style":494},[493],[385,7055,7057,7060],{"className":7056},[498],[385,7058,502],{"className":7059},[498],[385,7061,7063],{"className":7062},[506],[385,7064,7066,7086],{"className":7065},[510,511],[385,7067,7069,7083],{"className":7068},[515],[385,7070,7072],{"className":7071,"style":520},[519],[385,7073,7074,7077],{"style":523},[385,7075],{"className":7076,"style":528},[527],[385,7078,7080],{"className":7079},[532,533,534,535],[385,7081,433],{"className":7082,"style":539},[406,488,535],[385,7084,544],{"className":7085},[543],[385,7087,7089],{"className":7088},[515],[385,7090,7092],{"className":7091,"style":551},[519],[385,7093],{},[385,7095],{"className":7096,"style":494},[493],[385,7098,7100,7103],{"className":7099},[397],[385,7101],{"className":7102,"style":563},[401],[385,7104,4725],{"className":7105,"style":4724},[406,488]," Construct, from an\narbitrary instance of ",[385,7108,7110],{"className":7109},[388],[385,7111,7113],{"className":7112,"ariaHidden":393},[392],[385,7114,7116,7119],{"className":7115},[397],[385,7117],{"className":7118,"style":563},[401],[385,7120,4745],{"className":7121,"style":3180},[406,488],", an instance of ",[385,7124,7126],{"className":7125},[388],[385,7127,7129],{"className":7128,"ariaHidden":393},[392],[385,7130,7132,7135],{"className":7131},[397],[385,7133],{"className":7134,"style":563},[401],[385,7136,4725],{"className":7137,"style":4724},[406,488],". This is the creative heart of\nthe proof. ",[414,7140,7141],{},"Mind the direction:"," you transform ",[385,7144,7146],{"className":7145},[388],[385,7147,7149],{"className":7148,"ariaHidden":393},[392],[385,7150,7152,7155],{"className":7151},[397],[385,7153],{"className":7154,"style":563},[401],[385,7156,4745],{"className":7157,"style":3180},[406,488],"'s instance into ",[385,7160,7162],{"className":7161},[388],[385,7163,7165],{"className":7164,"ariaHidden":393},[392],[385,7166,7168,7171],{"className":7167},[397],[385,7169],{"className":7170,"style":563},[401],[385,7172,4725],{"className":7173,"style":4724},[406,488],"'s, so\nthat solving ",[385,7176,7178],{"className":7177},[388],[385,7179,7181],{"className":7180,"ariaHidden":393},[392],[385,7182,7184,7187],{"className":7183},[397],[385,7185],{"className":7186,"style":563},[401],[385,7188,4725],{"className":7189,"style":4724},[406,488]," would solve ",[385,7192,7194],{"className":7193},[388],[385,7195,7197],{"className":7196,"ariaHidden":393},[392],[385,7198,7200,7203],{"className":7199},[397],[385,7201],{"className":7202,"style":563},[401],[385,7204,4745],{"className":7205,"style":3180},[406,488],". Reducing the wrong way (",[385,7208,7210],{"className":7209},[388],[385,7211,7213,7268],{"className":7212,"ariaHidden":393},[392],[385,7214,7216,7219,7222,7225,7265],{"className":7215},[397],[385,7217],{"className":7218,"style":484},[401],[385,7220,4725],{"className":7221,"style":4724},[406,488],[385,7223],{"className":7224,"style":494},[493],[385,7226,7228,7231],{"className":7227},[498],[385,7229,502],{"className":7230},[498],[385,7232,7234],{"className":7233},[506],[385,7235,7237,7257],{"className":7236},[510,511],[385,7238,7240,7254],{"className":7239},[515],[385,7241,7243],{"className":7242,"style":520},[519],[385,7244,7245,7248],{"style":523},[385,7246],{"className":7247,"style":528},[527],[385,7249,7251],{"className":7250},[532,533,534,535],[385,7252,433],{"className":7253,"style":539},[406,488,535],[385,7255,544],{"className":7256},[543],[385,7258,7260],{"className":7259},[515],[385,7261,7263],{"className":7262,"style":551},[519],[385,7264],{},[385,7266],{"className":7267,"style":494},[493],[385,7269,7271,7274],{"className":7270},[397],[385,7272],{"className":7273,"style":563},[401],[385,7275,4745],{"className":7276,"style":3180},[406,488],") proves\nnothing about ",[385,7279,7281],{"className":7280},[388],[385,7282,7284],{"className":7283,"ariaHidden":393},[392],[385,7285,7287,7290],{"className":7286},[397],[385,7288],{"className":7289,"style":563},[401],[385,7291,4725],{"className":7292,"style":4724},[406,488],"'s hardness.",[1332,7295,7296,7299,7300,7302,7303,7305,7306,7321,7322,7305,7324,3492,7339,7341,7342,7305,7344,7359,7360,7362,7363,7378],{},[455,7297,7298],{},"Prove the reduction correct."," Establish the ",[414,7301,3365],{},": a\n",[455,7304,3255],{},"-instance of ",[385,7307,7309],{"className":7308},[388],[385,7310,7312],{"className":7311,"ariaHidden":393},[392],[385,7313,7315,7318],{"className":7314},[397],[385,7316],{"className":7317,"style":563},[401],[385,7319,4745],{"className":7320,"style":3180},[406,488]," maps to a ",[455,7323,3255],{},[385,7325,7327],{"className":7326},[388],[385,7328,7330],{"className":7329,"ariaHidden":393},[392],[385,7331,7333,7336],{"className":7332},[397],[385,7334],{"className":7335,"style":563},[401],[385,7337,4725],{"className":7338,"style":4724},[406,488],[455,7340,933],{},"\nconversely every ",[455,7343,3255],{},[385,7345,7347],{"className":7346},[388],[385,7348,7350],{"className":7349,"ariaHidden":393},[392],[385,7351,7353,7356],{"className":7352},[397],[385,7354],{"className":7355,"style":563},[401],[385,7357,4725],{"className":7358,"style":4724},[406,488]," comes only from a ",[455,7361,3255],{},"-instance\nof ",[385,7364,7366],{"className":7365},[388],[385,7367,7369],{"className":7368,"ariaHidden":393},[392],[385,7370,7372,7375],{"className":7371},[397],[385,7373],{"className":7374,"style":563},[401],[385,7376,4745],{"className":7377,"style":3180},[406,488],". Both directions are mandatory; a one-way implication leaves a hole\nthrough which false positives or negatives can slip.",[381,7380,7381,7382,7390,7391,7427,7428,7464],{},"Steps 1 and 2 are bookkeeping; steps 3 and 4 are where insight lives.",[1109,7383,7384],{},[460,7385,7389],{"href":7386,"ariaDescribedBy":7387,"dataFootnoteRef":376,"id":7388},"#user-content-fn-clrs-recipe",[1115],"user-content-fnref-clrs-recipe","5"," The\nworked reduction above is exactly this recipe applied with ",[385,7392,7394],{"className":7393},[388],[385,7395,7397,7415],{"className":7396,"ariaHidden":393},[392],[385,7398,7400,7403,7406,7409,7412],{"className":7399},[397],[385,7401],{"className":7402,"style":563},[401],[385,7404,4745],{"className":7405,"style":3180},[406,488],[385,7407],{"className":7408,"style":494},[493],[385,7410,1635],{"className":7411},[498],[385,7413],{"className":7414,"style":494},[493],[385,7416,7418,7421],{"className":7417},[397],[385,7419],{"className":7420,"style":4497},[401],[385,7422,7424],{"className":7423},[406,1567],[385,7425,4067],{"className":7426},[406,4444],"\nand ",[385,7429,7431],{"className":7430},[388],[385,7432,7434,7452],{"className":7433,"ariaHidden":393},[392],[385,7435,7437,7440,7443,7446,7449],{"className":7436},[397],[385,7438],{"className":7439,"style":563},[401],[385,7441,4725],{"className":7442,"style":4724},[406,488],[385,7444],{"className":7445,"style":494},[493],[385,7447,1635],{"className":7448},[498],[385,7450],{"className":7451,"style":494},[493],[385,7453,7455,7458],{"className":7454},[397],[385,7456],{"className":7457,"style":1563},[401],[385,7459,7461],{"className":7460},[406,1567],[385,7462,4867],{"className":7463},[406,4444],": the triangles-and-conflicts gadget is step\n3, and the two-direction argument is step 4.",[681,7466,7468],{"type":7467},"remark",[381,7469,7470,7473,7474,7489,7490,7492,7493,1088,7496,7511,7512,7515,7516,7519,7520,7535],{},[455,7471,7472],{},"Remark (The direction, once more)."," To prove ",[385,7475,7477],{"className":7476},[388],[385,7478,7480],{"className":7479,"ariaHidden":393},[392],[385,7481,7483,7486],{"className":7482},[397],[385,7484],{"className":7485,"style":563},[401],[385,7487,4725],{"className":7488,"style":4724},[406,488]," hard, reduce a ",[414,7491,6831],{}," problem\n",[455,7494,7495],{},"into",[385,7497,7499],{"className":7498},[388],[385,7500,7502],{"className":7501,"ariaHidden":393},[392],[385,7503,7505,7508],{"className":7504},[397],[385,7506],{"className":7507,"style":563},[401],[385,7509,4725],{"className":7510,"style":4724},[406,488],". The mantra: ",[414,7513,7514],{},"reduce from known-hard, to the new problem."," If you\never find yourself building an instance of a ",[414,7517,7518],{},"known"," problem out of ",[385,7521,7523],{"className":7522},[388],[385,7524,7526],{"className":7525,"ariaHidden":393},[392],[385,7527,7529,7532],{"className":7528},[397],[385,7530],{"className":7531,"style":563},[401],[385,7533,4725],{"className":7534,"style":4724},[406,488],", you\nhave the arrow backwards.",[465,7537,7539],{"id":7538},"takeaways","Takeaways",[1329,7541,7542,7643,7704,7738,7852],{},[1332,7543,7544,743,7559,7580,7581,7599,7600,7620,7621,7624,7642],{},[385,7545,7547],{"className":7546},[388],[385,7548,7550],{"className":7549,"ariaHidden":393},[392],[385,7551,7553,7556],{"className":7552},[397],[385,7554],{"className":7555,"style":563},[401],[385,7557,4725],{"className":7558,"style":4724},[406,488],[455,7560,7561,7579],{},[385,7562,7564],{"className":7563},[388],[385,7565,7567],{"className":7566,"ariaHidden":393},[392],[385,7568,7570,7573],{"className":7569},[397],[385,7571],{"className":7572,"style":402},[401],[385,7574,7576],{"className":7575},[406],[385,7577,411],{"className":7578},[406,410],"-hard"," if every problem in ",[385,7582,7584],{"className":7583},[388],[385,7585,7587],{"className":7586,"ariaHidden":393},[392],[385,7588,7590,7593],{"className":7589},[397],[385,7591],{"className":7592,"style":402},[401],[385,7594,7596],{"className":7595},[406],[385,7597,411],{"className":7598},[406,410]," reduces to it\n(a lower bound); it is ",[455,7601,7602,6760],{},[385,7603,7605],{"className":7604},[388],[385,7606,7608],{"className":7607,"ariaHidden":393},[392],[385,7609,7611,7614],{"className":7610},[397],[385,7612],{"className":7613,"style":402},[401],[385,7615,7617],{"className":7616},[406],[385,7618,411],{"className":7619},[406,410]," if it is also ",[414,7622,7623],{},"in",[385,7625,7627],{"className":7626},[388],[385,7628,7630],{"className":7629,"ariaHidden":393},[392],[385,7631,7633,7636],{"className":7632},[397],[385,7634],{"className":7635,"style":402},[401],[385,7637,7639],{"className":7638},[406],[385,7640,411],{"className":7641},[406,410],", hardest among the efficiently-checkable problems.",[1332,7644,1336,7645,7663,7664,7666,7667,7703],{},[385,7646,7648],{"className":7647},[388],[385,7649,7651],{"className":7650,"ariaHidden":393},[392],[385,7652,7654,7657],{"className":7653},[397],[385,7655],{"className":7656,"style":402},[401],[385,7658,7660],{"className":7659},[406],[385,7661,411],{"className":7662},[406,410],"-complete problems share one fate: a polynomial-time\nalgorithm for ",[455,7665,1410],{}," of them would prove ",[385,7668,7670],{"className":7669},[388],[385,7671,7673,7691],{"className":7672,"ariaHidden":393},[392],[385,7674,7676,7679,7682,7685,7688],{"className":7675},[397],[385,7677],{"className":7678,"style":402},[401],[385,7680,433],{"className":7681},[406,410],[385,7683],{"className":7684,"style":494},[493],[385,7686,1635],{"className":7687},[498],[385,7689],{"className":7690,"style":494},[493],[385,7692,7694,7697],{"className":7693},[397],[385,7695],{"className":7696,"style":402},[401],[385,7698,7700],{"className":7699},[406],[385,7701,411],{"className":7702},[406,410]," and solve\nthem all.",[1332,7705,7706,7709,7710,743,7712,7730,7731,7733,7734,7737],{},[455,7707,7708],{},"Cook–Levin"," anchors the theory: ",[455,7711,457],{},[385,7713,7715],{"className":7714},[388],[385,7716,7718],{"className":7717,"ariaHidden":393},[392],[385,7719,7721,7724],{"className":7720},[397],[385,7722],{"className":7723,"style":402},[401],[385,7725,7727],{"className":7726},[406],[385,7728,411],{"className":7729},[406,410],"-complete, proved\ndirectly by encoding any verifier's computation as a boolean formula. From it,\n",[455,7732,4067],{}," and a vast ",[455,7735,7736],{},"reduction web"," follow by transitivity.",[1332,7739,7740,7741,7817,7818,7851],{},"The ",[385,7742,7744],{"className":7743},[388],[385,7745,7747,7805],{"className":7746,"ariaHidden":393},[392],[385,7748,7750,7753,7759,7762,7802],{"className":7749},[397],[385,7751],{"className":7752,"style":4437},[401],[385,7754,7756],{"className":7755},[406,1567],[385,7757,4067],{"className":7758},[406,4444],[385,7760],{"className":7761,"style":494},[493],[385,7763,7765,7768],{"className":7764},[498],[385,7766,502],{"className":7767},[498],[385,7769,7771],{"className":7770},[506],[385,7772,7774,7794],{"className":7773},[510,511],[385,7775,7777,7791],{"className":7776},[515],[385,7778,7780],{"className":7779,"style":520},[519],[385,7781,7782,7785],{"style":523},[385,7783],{"className":7784,"style":528},[527],[385,7786,7788],{"className":7787},[532,533,534,535],[385,7789,433],{"className":7790,"style":539},[406,488,535],[385,7792,544],{"className":7793},[543],[385,7795,7797],{"className":7796},[515],[385,7798,7800],{"className":7799,"style":551},[519],[385,7801],{},[385,7803],{"className":7804,"style":494},[493],[385,7806,7808,7811],{"className":7807},[397],[385,7809],{"className":7810,"style":1563},[401],[385,7812,7814],{"className":7813},[406,1567],[385,7815,4867],{"className":7816},[406,4444]," reduction (a triangle per\nclause plus conflict edges, with ",[385,7819,7821],{"className":7820},[388],[385,7822,7824,7842],{"className":7823,"ariaHidden":393},[392],[385,7825,7827,7830,7833,7836,7839],{"className":7826},[397],[385,7828],{"className":7829,"style":402},[401],[385,7831,5024],{"className":7832,"style":5023},[406,488],[385,7834],{"className":7835,"style":494},[493],[385,7837,1635],{"className":7838},[498],[385,7840],{"className":7841,"style":494},[493],[385,7843,7845,7848],{"className":7844},[397],[385,7846],{"className":7847,"style":2961},[401],[385,7849,4989],{"className":7850},[406,488],") is the model gadget reduction.",[1332,7853,7854,7855,7873,7874,7877,7878,3492,7896,7899,7900,2754,7918,3492,7933,7936,7937,8007,8008,8011,8012,8014],{},"To prove a new problem ",[385,7856,7858],{"className":7857},[388],[385,7859,7861],{"className":7860,"ariaHidden":393},[392],[385,7862,7864,7867],{"className":7863},[397],[385,7865],{"className":7866,"style":402},[401],[385,7868,7870],{"className":7869},[406],[385,7871,411],{"className":7872},[406,410],"-complete: ",[455,7875,7876],{},"(1)"," show membership in\n",[385,7879,7881],{"className":7880},[388],[385,7882,7884],{"className":7883,"ariaHidden":393},[392],[385,7885,7887,7890],{"className":7886},[397],[385,7888],{"className":7889,"style":402},[401],[385,7891,7893],{"className":7892},[406],[385,7894,411],{"className":7895},[406,410],[455,7897,7898],{},"(2)"," pick a known ",[385,7901,7903],{"className":7902},[388],[385,7904,7906],{"className":7905,"ariaHidden":393},[392],[385,7907,7909,7912],{"className":7908},[397],[385,7910],{"className":7911,"style":402},[401],[385,7913,7915],{"className":7914},[406],[385,7916,411],{"className":7917},[406,410],[385,7919,7921],{"className":7920},[388],[385,7922,7924],{"className":7923,"ariaHidden":393},[392],[385,7925,7927,7930],{"className":7926},[397],[385,7928],{"className":7929,"style":563},[401],[385,7931,4745],{"className":7932,"style":3180},[406,488],[455,7934,7935],{},"(3)"," reduce\n",[385,7938,7940],{"className":7939},[388],[385,7941,7943,7998],{"className":7942,"ariaHidden":393},[392],[385,7944,7946,7949,7952,7955,7995],{"className":7945},[397],[385,7947],{"className":7948,"style":484},[401],[385,7950,4745],{"className":7951,"style":3180},[406,488],[385,7953],{"className":7954,"style":494},[493],[385,7956,7958,7961],{"className":7957},[498],[385,7959,502],{"className":7960},[498],[385,7962,7964],{"className":7963},[506],[385,7965,7967,7987],{"className":7966},[510,511],[385,7968,7970,7984],{"className":7969},[515],[385,7971,7973],{"className":7972,"style":520},[519],[385,7974,7975,7978],{"style":523},[385,7976],{"className":7977,"style":528},[527],[385,7979,7981],{"className":7980},[532,533,534,535],[385,7982,433],{"className":7983,"style":539},[406,488,535],[385,7985,544],{"className":7986},[543],[385,7988,7990],{"className":7989},[515],[385,7991,7993],{"className":7992,"style":551},[519],[385,7994],{},[385,7996],{"className":7997,"style":494},[493],[385,7999,8001,8004],{"className":8000},[397],[385,8002],{"className":8003,"style":563},[401],[385,8005,4725],{"className":8006,"style":4724},[406,488]," in polynomial time, ",[455,8009,8010],{},"(4)"," prove the equivalence both ways,\nalways reducing ",[414,8013,6897],{}," the hard problem.",[8016,8017,8020,8025],"section",{"className":8018,"dataFootnotes":376},[8019],"footnotes",[465,8021,8024],{"className":8022,"id":1115},[8023],"sr-only","Footnotes",[6763,8026,8027,8098,8130,8161,8283],{},[1332,8028,8030,8033,8034,8052,8053,8071,8072,8090,8091],{"id":8029},"user-content-fn-clrs-nphard",[455,8031,8032],{},"CLRS",", Ch. 34 — NP-Completeness (§34.1): the definitions of ",[385,8035,8037],{"className":8036},[388],[385,8038,8040],{"className":8039,"ariaHidden":393},[392],[385,8041,8043,8046],{"className":8042},[397],[385,8044],{"className":8045,"style":402},[401],[385,8047,8049],{"className":8048},[406],[385,8050,411],{"className":8051},[406,410],"-hard (a lower bound) and ",[385,8054,8056],{"className":8055},[388],[385,8057,8059],{"className":8058,"ariaHidden":393},[392],[385,8060,8062,8065],{"className":8061},[397],[385,8063],{"className":8064,"style":402},[401],[385,8066,8068],{"className":8067},[406],[385,8069,411],{"className":8070},[406,410],"-complete (hard and in ",[385,8073,8075],{"className":8074},[388],[385,8076,8078],{"className":8077,"ariaHidden":393},[392],[385,8079,8081,8084],{"className":8080},[397],[385,8082],{"className":8083,"style":402},[401],[385,8085,8087],{"className":8086},[406],[385,8088,411],{"className":8089},[406,410],"). ",[460,8092,8097],{"href":8093,"ariaLabel":8094,"className":8095,"dataFootnoteBackref":376},"#user-content-fnref-clrs-nphard","Back to reference 1",[8096],"data-footnote-backref","↩",[1332,8099,8101,8103,8104,743,8106,8124,8125],{"id":8100},"user-content-fn-clrs-cook",[455,8102,8032],{},", Ch. 34 — NP-Completeness (§34.3): the Cook–Levin theorem that ",[455,8105,457],{},[385,8107,8109],{"className":8108},[388],[385,8110,8112],{"className":8111,"ariaHidden":393},[392],[385,8113,8115,8118],{"className":8114},[397],[385,8116],{"className":8117,"style":402},[401],[385,8119,8121],{"className":8120},[406],[385,8122,411],{"className":8123},[406,410],"-complete, proved by encoding a verifier's computation as a boolean formula. ",[460,8126,8097],{"href":8127,"ariaLabel":8128,"className":8129,"dataFootnoteBackref":376},"#user-content-fnref-clrs-cook","Back to reference 2",[8096],[1332,8131,8133,8136,8137,8155,8156],{"id":8132},"user-content-fn-skiena-web",[455,8134,8135],{},"Skiena",", §11 — NP-Completeness: Karp's 1972 reductions and the web of ",[385,8138,8140],{"className":8139},[388],[385,8141,8143],{"className":8142,"ariaHidden":393},[392],[385,8144,8146,8149],{"className":8145},[397],[385,8147],{"className":8148,"style":402},[401],[385,8150,8152],{"className":8151},[406],[385,8153,411],{"className":8154},[406,410],"-complete problems growing from satisfiability. ",[460,8157,8097],{"href":8158,"ariaLabel":8159,"className":8160,"dataFootnoteBackref":376},"#user-content-fnref-skiena-web","Back to reference 3",[8096],[1332,8162,8164,8167,8168,8244,8245,8090,8278],{"id":8163},"user-content-fn-erickson-gadget",[455,8165,8166],{},"Erickson",", Ch. 12 — NP-Hardness: the gadget reduction ",[385,8169,8171],{"className":8170},[388],[385,8172,8174,8232],{"className":8173,"ariaHidden":393},[392],[385,8175,8177,8180,8186,8189,8229],{"className":8176},[397],[385,8178],{"className":8179,"style":4437},[401],[385,8181,8183],{"className":8182},[406,1567],[385,8184,4067],{"className":8185},[406,4444],[385,8187],{"className":8188,"style":494},[493],[385,8190,8192,8195],{"className":8191},[498],[385,8193,502],{"className":8194},[498],[385,8196,8198],{"className":8197},[506],[385,8199,8201,8221],{"className":8200},[510,511],[385,8202,8204,8218],{"className":8203},[515],[385,8205,8207],{"className":8206,"style":520},[519],[385,8208,8209,8212],{"style":523},[385,8210],{"className":8211,"style":528},[527],[385,8213,8215],{"className":8214},[532,533,534,535],[385,8216,433],{"className":8217,"style":539},[406,488,535],[385,8219,544],{"className":8220},[543],[385,8222,8224],{"className":8223},[515],[385,8225,8227],{"className":8226,"style":551},[519],[385,8228],{},[385,8230],{"className":8231,"style":494},[493],[385,8233,8235,8238],{"className":8234},[397],[385,8236],{"className":8237,"style":1563},[401],[385,8239,8241],{"className":8240},[406,1567],[385,8242,4867],{"className":8243},[406,4444]," (clause triangles plus conflict edges, ",[385,8246,8248],{"className":8247},[388],[385,8249,8251,8269],{"className":8250,"ariaHidden":393},[392],[385,8252,8254,8257,8260,8263,8266],{"className":8253},[397],[385,8255],{"className":8256,"style":402},[401],[385,8258,5024],{"className":8259,"style":5023},[406,488],[385,8261],{"className":8262,"style":494},[493],[385,8264,1635],{"className":8265},[498],[385,8267],{"className":8268,"style":494},[493],[385,8270,8272,8275],{"className":8271},[397],[385,8273],{"className":8274,"style":2961},[401],[385,8276,4989],{"className":8277},[406,488],[460,8279,8097],{"href":8280,"ariaLabel":8281,"className":8282,"dataFootnoteBackref":376},"#user-content-fnref-erickson-gadget","Back to reference 4",[8096],[1332,8284,8286,8288,8289,8307,8308],{"id":8285},"user-content-fn-clrs-recipe",[455,8287,8032],{},", Ch. 34 — NP-Completeness (§34.4): the standard four-step recipe for proving a new problem ",[385,8290,8292],{"className":8291},[388],[385,8293,8295],{"className":8294,"ariaHidden":393},[392],[385,8296,8298,8301],{"className":8297},[397],[385,8299],{"className":8300,"style":402},[401],[385,8302,8304],{"className":8303},[406],[385,8305,411],{"className":8306},[406,410],"-complete. ",[460,8309,8097],{"href":8310,"ariaLabel":8311,"className":8312,"dataFootnoteBackref":376},"#user-content-fnref-clrs-recipe","Back to reference 5",[8096],{"title":376,"searchDepth":18,"depth":18,"links":8314},[8315,8316,8317,8320,8321,8322],{"id":467,"depth":18,"text":468},{"id":2731,"depth":18,"text":2732},{"id":4029,"depth":18,"text":4030,"children":8318},[8319],{"id":4876,"depth":24,"text":4877},{"id":6682,"depth":18,"text":6683},{"id":7538,"depth":18,"text":7539},{"id":1115,"depth":18,"text":8024},"The previous lesson left us with a claim: inside NP\nthere are problems that are universally hardest, and they hold the key to the\nP versus NP question. This lesson makes that claim\nprecise. We define what it means to be hardest, identify the first such problem\n(SAT, via the Cook–Levin theorem), and then show how a single seed problem\nsprouts an entire forest of equally hard problems through reductions. Finally\nwe distill the whole enterprise into a four-step recipe you can apply to a\nproblem you have never seen before.","md",{"moduleNumber":196,"lessonNumber":18,"order":8326},802,true,[8329,8333,8337,8340],{"title":8330,"slug":8331,"difficulty":8332},"Flower Planting With No Adjacent","flower-planting-with-no-adjacent","Medium",{"title":8334,"slug":8335,"difficulty":8336},"N-Queens","n-queens","Hard",{"title":8338,"slug":8339,"difficulty":8336},"Sudoku Solver","sudoku-solver",{"title":8341,"slug":8342,"difficulty":8336},"Find the Shortest Superstring","find-the-shortest-superstring","---\ntitle: NP-Completeness\nmodule: Intractability\nmoduleNumber: 8\nlessonNumber: 2\norder: 802\nsummary: >-\n  Some problems in $\\mathsf{NP}$ are universally hardest: every other problem\n  in $\\mathsf{NP}$ reduces to them. This lesson defines $\\mathsf{NP}$-hard and\n  $\\mathsf{NP}$-complete, states the Cook–Levin theorem that anchors the whole\n  edifice on **SAT**, walks the web of reductions that grows from it, and gives\n  the four-step recipe for proving a brand-new problem $\\mathsf{NP}$-complete.\ntopics: [NP-Completeness]\nsources:\n  - book: CLRS\n    ref: \"Ch. 34 — NP-Completeness\"\n  - book: Skiena\n    ref: \"§11 — NP-Completeness\"\n  - book: Erickson\n    ref: \"Ch. 12 — NP-Hardness\"\npractice:\n  - title: 'Flower Planting With No Adjacent'\n    slug: flower-planting-with-no-adjacent\n    difficulty: Medium\n  - title: 'N-Queens'\n    slug: n-queens\n    difficulty: Hard\n  - title: 'Sudoku Solver'\n    slug: sudoku-solver\n    difficulty: Hard\n  - title: 'Find the Shortest Superstring'\n    slug: find-the-shortest-superstring\n    difficulty: Hard\n---\n\nThe previous lesson left us with a claim: inside $\\mathsf{NP}$\nthere are problems that are _universally hardest_, and they hold the key to the\n$\\mathsf{P}$ versus $\\mathsf{NP}$ question. This lesson makes that claim\nprecise. We define what it means to be hardest, identify the first such problem\n(**SAT**, via the Cook–Levin theorem), and then show how a single seed problem\nsprouts an entire forest of equally hard problems through [reductions](\u002Falgorithms\u002Fintractability\u002Fp-np-reductions). Finally\nwe distill the whole enterprise into a four-step recipe you can apply to a\nproblem you have never seen before.\n\n## NP-hard and NP-complete\n\nRecall that $A \\le_P B$ means \"$A$ is no harder than $B$.\" Now imagine a\nproblem $B$ that _every_ problem in $\\mathsf{NP}$ is no harder than. Such a $B$\nis at least as hard as everything in $\\mathsf{NP}$, a ceiling on the whole\nclass.\n\n> **Definition ($\\mathsf{NP}$-hard and $\\mathsf{NP}$-complete).** A problem $B$ is $\\mathsf{NP}$-hard if $A \\le_P B$ for _every_ problem $A\n> \\in \\mathsf{NP}$.\n>\n> A problem $B$ is $\\mathsf{NP}$-complete if it is $\\mathsf{NP}$-hard _and_\n> $B \\in \\mathsf{NP}$.\n\nThe distinction matters. $\\mathsf{NP}$-hardness is a pure _lower bound_: $B$ is\nas hard as anything in $\\mathsf{NP}$, but $B$ need not itself be in\n$\\mathsf{NP}$; it could be far harder, even undecidable. $\\mathsf{NP}$-complete\nproblems are the ones that are hardest _and still belong to_ $\\mathsf{NP}$:\nthey sit exactly at the frontier. They are the hardest\nproblems whose solutions we can still efficiently check.[^clrs-nphard]\n\n$$\n% caption: $\\mathsf{NP}$-complete is the intersection of $\\mathsf{NP}$ and\n%          $\\mathsf{NP}$-hard; some $\\mathsf{NP}$-hard problems lie outside $\\mathsf{NP}$.\n\\begin{tikzpicture}[font=\\small]\n  \\definecolor{acc}{HTML}{2348F2}\n  % NP ellipse (left)\n  \\draw[thick] (0,0) ellipse (2.5 and 1.7);\n  \\node[anchor=south] at (-1.4,1.05) {$\\mathsf{NP}$};\n  % NP-hard ellipse (right), overlapping\n  \\draw[thick] (3.0,0) ellipse (2.5 and 1.7);\n  \\node[anchor=south] at (4.4,1.05) {$\\mathsf{NP}$-hard};\n  % P region inside NP only\n  \\draw[draw=acc, fill=acc!15] (-1.5,-0.2) ellipse (0.75 and 0.5);\n  \\node[font=\\scriptsize] at (-1.5,-0.2) {$\\mathsf{P}$};\n  % intersection label\n  \\node[align=center, font=\\scriptsize, fill=acc!15, draw=acc, very thick,\n        rounded corners, inner sep=2pt] at (1.5,0)\n    {$\\mathsf{NP}$-\\\\complete};\n  % outside-NP hard example\n  \\node[font=\\scriptsize, align=center] at (4.3,-0.55) {halting\\\\(undecidable)};\n\\end{tikzpicture}\n$$\n\nTwo consequences make these definitions powerful.\n\n- **All $\\mathsf{NP}$-complete problems stand or fall together.** Suppose $B$ is\n  $\\mathsf{NP}$-complete and someone finds a polynomial-time algorithm for $B$.\n  Then for _any_ $A \\in \\mathsf{NP}$ we have $A \\le_P B$, so $A$ is solvable in\n  polynomial time too, every $A$ at once. Hence\n  $$ \\text{any } \\mathsf{NP}\\text{-complete problem} \\in \\mathsf{P} \\;\\Longrightarrow\\; \\mathsf{P} = \\mathsf{NP}. $$\n  Conversely, proving _any single_ $\\mathsf{NP}$-complete problem requires\n  super-polynomial time would prove $\\mathsf{P} \\ne \\mathsf{NP}$. In this sense\n  the thousands of known $\\mathsf{NP}$-complete problems are a single problem\n  in many guises.\n- **Transitivity bootstraps the class.** If $C$ is $\\mathsf{NP}$-complete and we\n  show $C \\le_P D$ for some $D \\in \\mathsf{NP}$, then $D$ is\n  $\\mathsf{NP}$-complete too: every $A \\in \\mathsf{NP}$ satisfies $A \\le_P C\n  \\le_P D$, and $\\le_P$ composes. This is the engine that turns _one_\n  $\\mathsf{NP}$-complete problem into many.\n\n$$\n% caption: Transitivity bootstrap: every $A\\in\\mathsf{NP}$ already reduces to the\n%          known-complete $C$; one new reduction $C\\le_P D$ extends the chain, so $D$\n%          inherits hardness from the whole class.\n\\begin{tikzpicture}[>=Stealth, font=\\small,\n    p\u002F.style={draw, rounded corners, minimum height=9mm, minimum width=20mm, align=center}]\n  \\definecolor{acc}{HTML}{2348F2}\n  % the whole class NP fans in to C\n  \\node (a1) at (0,1.5) {$A_1$};\n  \\node (a2) at (0,0.75) {$A_2$};\n  \\node[font=\\scriptsize] (dots) at (0,0.1) {$\\vdots$};\n  \\node (ak) at (0,-0.6) {$A_k$};\n  \\node[font=\\scriptsize, anchor=east] at (-0.45,0.45) {every $A\\in\\mathsf{NP}$};\n  \\node[p, draw=acc, fill=acc!15] (c) at (4.2,0.45) {$C$\\\\(known complete)};\n  \\node[p, draw=acc, very thick] (d) at (8.6,0.45) {$D$\\\\(new, in $\\mathsf{NP}$)};\n  \\draw[->] (a1) -- (c.165);\n  \\draw[->] (a2) -- (c.178);\n  \\draw[->] (ak) -- (c.192);\n  \\draw[->, acc, thick] (c) -- node[above, font=\\scriptsize]{$C\\le_P D$} (d);\n  % the bootstrapped chain, as a flat leader beneath the row\n  \\draw[acc] (-0.4,-1.15) -- (8.6,-1.15);\n  \\draw[acc] (-0.4,-1.05) -- (-0.4,-1.25);\n  \\draw[acc] (8.6,-1.05) -- (8.6,-1.25);\n  \\node[below, font=\\scriptsize, text=acc, align=center] at (4.1,-1.25)\n    {so $A\\le_P C\\le_P D$ for every $A$\\, $\\Rightarrow$\\, $D$ is $\\mathsf{NP}$-complete};\n\\end{tikzpicture}\n$$\n\nBut the engine needs a spark. Transitivity spreads\n$\\mathsf{NP}$-completeness from one problem to the next, yet it presupposes we\nalready have a _first_ $\\mathsf{NP}$-complete problem to start from. Where does\nthe first one come from?\n\n## The first one: Cook–Levin\n\nThe breakthrough, proved independently by Stephen Cook (1971) and Leonid Levin,\nis that a natural problem is $\\mathsf{NP}$-complete _from scratch_, without\nreducing from anything, by reasoning directly about computation itself.\n\nThe problem is **boolean satisfiability**, or **SAT**.\n\n> **Definition (SAT).** Given a boolean formula $\\varphi$ over variables $x_1, \\dots, x_n$\n> using $\\wedge$ (and), $\\vee$ (or), and $\\neg$ (not), is there an assignment of\n> **true**\u002F**false** to the variables that makes $\\varphi$ evaluate to **true**?\n\n> **Theorem (Cook–Levin).** **SAT** is $\\mathsf{NP}$-complete.\n\nThat **SAT** is in $\\mathsf{NP}$ is easy: a satisfying assignment is a\ncertificate, and evaluating $\\varphi$ on it takes linear time. The deep half is\n$\\mathsf{NP}$-hardness: showing that _every_ $A \\in \\mathsf{NP}$ reduces to\n**SAT**. The idea, which we sketch rather than prove, is to encode _computation\nas logic_. Any $A \\in \\mathsf{NP}$ has a polynomial-time verifier $V$. For an\ninput $x$, the question \"does some certificate make $V$ accept $x$?\" is exactly\nthe question \"is $A$'s answer **yes**?\" Cook and Levin build, in polynomial\ntime, a boolean formula $\\varphi_x$ whose variables describe the verifier's\nentire step-by-step execution, with clauses that force the variables to obey\nthe machine's rules. Then $\\varphi_x$ is satisfiable **if and only if** some\ncertificate makes $V$ accept, so deciding **SAT** on $\\varphi_x$ decides $A$\non $x$. Because this works for an _arbitrary_ problem in $\\mathsf{NP}$, **SAT**\nis $\\mathsf{NP}$-hard.[^clrs-cook]\n\n$$\n% caption: Cook–Levin encodes a verifier's whole computation on $(x,y)$ as one formula\n%          $\\varphi_x$, satisfiable iff some $y$ accepts.\n\\begin{tikzpicture}[>=Stealth, font=\\small,\n    box\u002F.style={draw, rectangle, rounded corners, minimum height=13mm,\n                minimum width=22mm, align=center}]\n  \\definecolor{acc}{HTML}{2348F2}\n  \\node[box] (x)   at (0,0)    {input $x$\\\\(fixed)};\n  \\node[box] (v)   at (3.6,0)  {verifier $V$\\\\runs poly steps};\n  \\node[box, draw=acc] (phi) at (8.0,0) {formula $\\varphi_x$\\\\(built in poly time)};\n  \\node[box] (sat) at (12.2,0) {SAT\\\\on $\\varphi_x$};\n  \\draw[->] (x) -- (v);\n  \\draw[->] (v) -- (phi);\n  \\draw[->, acc] (phi) -- (sat);\n  % edge label lifted clear of both boxes, tied to the arrow by a short leader\n  \\node[font=\\scriptsize, text=red!75!black] (enc) at (5.8,1.15) {encode as logic};\n  \\draw[->, red!75!black, shorten >=1pt] (enc.south) -- (5.8,0.16);\n  \\node[anchor=north, align=center, font=\\scriptsize] at (3.6,-0.75)\n    {variables = tape\\\\+ state at each step};\n  \\node[anchor=north, align=center, font=\\scriptsize] at (8.0,-0.75)\n    {clauses force\\\\legal transitions};\n  \\node[anchor=north, align=center, font=\\scriptsize] at (12.2,-0.75)\n    {satisfiable $\\iff$\\\\some $y$ accepts};\n\\end{tikzpicture}\n$$\n\nWith one $\\mathsf{NP}$-complete problem in hand, the transitivity engine takes\nover.\n\n## The reduction web\n\nKarp's celebrated 1972 paper reduced **SAT** to twenty-one other problems,\nshowing them all $\\mathsf{NP}$-complete and launching the field.[^skiena-web] A convenient\nwaystation is **3-SAT**, the special case of **SAT** in which the formula is in\n_conjunctive normal form_ with exactly three literals per clause, a big AND of\nsmall ORs such as $(x_1 \\vee \\neg x_2 \\vee x_3) \\wedge (\\neg x_1 \\vee x_2 \\vee\nx_4)$. One can show $\\textbf{SAT} \\le_P \\textbf{3-SAT}$, so **3-SAT** is itself\n$\\mathsf{NP}$-complete, and its rigid structure makes it the favorite\nstarting point for further reductions.\n\nFrom **3-SAT** the web branches out. A small portion:\n\n$$\n% caption: Reduction web of NP-complete problems branching out from SAT and 3-SAT.\n\\begin{tikzpicture}[>=Stealth,\n    p\u002F.style={draw, rounded corners, minimum height=10mm, inner sep=4pt}]\n  \\node (sat)  [p] at (0,0)      {\\textbf{SAT}};\n  \\node (3sat) [p] at (2.7,0)    {\\textbf{3-SAT}};\n  \\node (is)   [p] at (5.9,1.4)  {\\textbf{Independent-Set}};\n  \\node (ss)   [p] at (5.9,-1.4) {\\textbf{Subset-Sum}};\n  \\node (clq)  [p] at (9.9,2.6)  {\\textbf{Clique}};\n  \\node (vc)   [p] at (9.9,1.4)  {\\textbf{Vertex-Cover}};\n  \\node (ham)  [p] at (9.9,0.2)  {\\textbf{Ham-Cycle}};\n  \\draw[->] (sat) -- (3sat);\n  \\draw[->] (3sat) -- (is);\n  \\draw[->] (3sat) -- (ss);\n  \\draw[->] (is) -- (clq);\n  \\draw[->] (is) -- (vc);\n  \\draw[->] (vc) -- (ham);\n\\end{tikzpicture}\n$$\n\nEvery arrow $X \\to Y$ is a reduction $X \\le_P Y$; following arrows back to\n**SAT** certifies each box as $\\mathsf{NP}$-complete. (The arrows above record\none _route_ to each result, not the only one; many of these problems also\nreduce to each other directly, as we saw with $\\textsc{Independent-Set}$ and\n**Clique** last lesson.)\n\n### A worked reduction: 3-SAT to Independent-Set\n\nLet us actually build one arrow, the classic $\\textbf{3-SAT} \\le_P\n\\textbf{Independent-Set}$. We are given a 3-CNF formula $\\varphi$ with $m$\nclauses and must produce a graph $G$ and integer $k$ such that $G$ has an\n[independent set](\u002Falgorithms\u002Fgraphs\u002Frepresentations-and-traversal) of size $k$ exactly when $\\varphi$ is satisfiable.\n\n**The construction.** For each clause, create a _triangle_ of three vertices,\none per literal. So clause $(x_1 \\vee \\neg x_2 \\vee x_3)$ becomes three\nmutually-connected vertices labeled $x_1$, $\\neg x_2$, $x_3$. Then add a\n**conflict edge** between any two vertices in different triangles that hold\n_contradictory_ literals: one labeled $x_i$ and another labeled $\\neg x_i$.\nFinally set $k = m$, the number of clauses. This is clearly polynomial: $3m$\nvertices and at most $O(m^2)$ edges.\n\n**Why it works.** An independent set may pick _at most one_ vertex from each\ntriangle (the three are mutually adjacent), so an independent set of size $k =\nm$ must pick _exactly one_ literal from every clause. The conflict edges forbid\nchoosing both $x_i$ and $\\neg x_i$ anywhere, so the chosen literals are\nmutually consistent — they describe a partial truth assignment. Setting each\nchosen literal **true** satisfies one literal in every clause, hence satisfies\n$\\varphi$. Run the argument backward: a satisfying assignment picks, from each\nclause, one true literal; those $m$ vertices form an independent set, since two\ntrue literals can never be a variable and its negation. Therefore\n$$ \\varphi \\text{ is satisfiable} \\iff G \\text{ has an independent set of size } m, $$\nwhich is exactly what a valid reduction requires.[^erickson-gadget]\n\n$$\n% caption: Clause triangles with dashed conflict edges for the 3-SAT to Independent-Set\n%          reduction.\n\\begin{tikzpicture}[>=Stealth,\n    v\u002F.style={draw, circle, inner sep=1.5pt, minimum size=6mm, font=\\small}]\n  % clause 1 triangle\n  \\node[v] (a1) at (0,1.2) {$x_1$};\n  \\node[v] (a2) at (-0.7,0) {$\\neg x_2$};\n  \\node[v] (a3) at (0.7,0) {$x_3$};\n  \\draw (a1)--(a2)--(a3)--(a1);\n  % clause 2 triangle\n  \\node[v] (b1) at (4,1.2) {$\\neg x_1$};\n  \\node[v] (b2) at (3.3,0) {$x_2$};\n  \\node[v] (b3) at (4.7,0) {$x_4$};\n  \\draw (b1)--(b2)--(b3)--(b1);\n  % conflict edges\n  \\draw[dashed] (a1) to[bend left=10] (b1);\n  \\draw[dashed] (a2) to[bend right=18] (b2);\n  \\node[font=\\small] at (2,1.55) {conflict};\n\\end{tikzpicture}\n$$\n\nSolid edges are the per-clause triangles; dashed edges connect contradictory\nliterals ($x_1$ vs. $\\neg x_1$, and $\\neg x_2$ vs. $x_2$). Picking one\nnon-conflicting vertex per triangle is exactly a consistent satisfying choice.\n\n$$\n% caption: Choosing $\\{x_1, x_2\\}$ — one vertex per triangle, no conflict edge between\n%          them — is a size-$2$ independent set, i.e. a satisfying assignment.\n\\begin{tikzpicture}[>=Stealth, font=\\small,\n    v\u002F.style={draw, circle, inner sep=1.5pt, minimum size=6mm},\n    sel\u002F.style={draw=acc, circle, fill=acc!15, very thick, inner sep=1.5pt, minimum size=6mm}]\n  \\definecolor{acc}{HTML}{2348F2}\n  % clause 1 triangle: pick x_1\n  \\node[sel] (a1) at (0,1.2) {$x_1$};\n  \\node[v] (a2) at (-0.7,0) {$\\neg x_2$};\n  \\node[v] (a3) at (0.7,0) {$x_3$};\n  \\draw (a1)--(a2)--(a3)--(a1);\n  % clause 2 triangle: pick x_2\n  \\node[v] (b1) at (4,1.2) {$\\neg x_1$};\n  \\node[sel] (b2) at (3.3,0) {$x_2$};\n  \\node[v] (b3) at (4.7,0) {$x_4$};\n  \\draw (b1)--(b2)--(b3)--(b1);\n  % conflict edges (dashed)\n  \\draw[dashed] (a1) to[bend left=10] (b1);\n  \\draw[dashed] (a2) to[bend right=18] (b2);\n  \\node[font=\\scriptsize] at (2,1.55) {conflict};\n  \\node[anchor=north, align=center, font=\\scriptsize, text=acc] at (2,-0.5)\n    {chosen set $\\{x_1,x_2\\}$: independent $\\Rightarrow$ $x_1=x_2=\\text{true}$ satisfies $\\varphi$};\n\\end{tikzpicture}\n$$\n\n## The recipe: proving a new problem NP-complete\n\nOnce a stockpile of $\\mathsf{NP}$-complete problems exists, classifying a\n_new_ problem $X$ follows a fixed procedure. To prove **$X$ is\n$\\mathsf{NP}$-complete**, carry out four steps.\n\n1. **Show $X \\in \\mathsf{NP}$.** Describe a polynomial-size certificate for\n   **yes**-instances and argue it can be checked in polynomial time. This is\n   usually the easy step, but skipping it is a real error: an\n   $\\mathsf{NP}$-_hard_ problem outside $\\mathsf{NP}$ is hard but not\n   _complete_.\n2. **Choose a known $\\mathsf{NP}$-complete problem $Y$** to reduce _from_.\n   Pick one whose structure resembles $X$ — **3-SAT** for logical or\n   gadget-style constraints, $\\textsc{Vertex-Cover}$ or $\\textsc{Independent-Set}$ for graph\n   selection, $\\textsc{Subset-Sum}$ or $\\textsc{Partition}$ for numeric targets,\n   $\\textsc{Hamiltonian-Cycle}$ for routing.\n3. **Give a polynomial-time reduction $Y \\le_P X$.** Construct, from an\n   arbitrary instance of $Y$, an instance of $X$. This is the creative heart of\n   the proof. _Mind the direction:_ you transform $Y$'s instance into $X$'s, so\n   that solving $X$ would solve $Y$. Reducing the wrong way ($X \\le_P Y$) proves\n   nothing about $X$'s hardness.\n4. **Prove the reduction correct.** Establish the _if and only if_: a\n   **yes**-instance of $Y$ maps to a **yes**-instance of $X$, **and**\n   conversely every **yes**-instance of $X$ comes only from a **yes**-instance\n   of $Y$. Both directions are mandatory; a one-way implication leaves a hole\n   through which false positives or negatives can slip.\n\nSteps 1 and 2 are bookkeeping; steps 3 and 4 are where insight lives.[^clrs-recipe] The\nworked reduction above is exactly this recipe applied with $Y = \\textbf{3-SAT}$\nand $X = \\textbf{Independent-Set}$: the triangles-and-conflicts gadget is step\n3, and the two-direction argument is step 4.\n\n> **Remark (The direction, once more).** To prove $X$ hard, reduce a _hard_ problem\n> **into** $X$. The mantra: _reduce from known-hard, to the new problem._ If you\n> ever find yourself building an instance of a _known_ problem out of $X$, you\n> have the arrow backwards.\n\n## Takeaways\n\n- $X$ is **$\\mathsf{NP}$-hard** if every problem in $\\mathsf{NP}$ reduces to it\n  (a lower bound); it is **$\\mathsf{NP}$-complete** if it is also _in_\n  $\\mathsf{NP}$, hardest among the efficiently-checkable problems.\n- All $\\mathsf{NP}$-complete problems share one fate: a polynomial-time\n  algorithm for **any** of them would prove $\\mathsf{P} = \\mathsf{NP}$ and solve\n  them all.\n- **Cook–Levin** anchors the theory: **SAT** is $\\mathsf{NP}$-complete, proved\n  directly by encoding any verifier's computation as a boolean formula. From it,\n  **3-SAT** and a vast **reduction web** follow by transitivity.\n- The $\\textbf{3-SAT} \\le_P \\textbf{Independent-Set}$ reduction (a triangle per\n  clause plus conflict edges, with $k = m$) is the model gadget reduction.\n- To prove a new problem $\\mathsf{NP}$-complete: **(1)** show membership in\n  $\\mathsf{NP}$, **(2)** pick a known $\\mathsf{NP}$-complete $Y$, **(3)** reduce\n  $Y \\le_P X$ in polynomial time, **(4)** prove the equivalence both ways,\n  always reducing _from_ the hard problem.\n\n[^clrs-nphard]: **CLRS**, Ch. 34 — NP-Completeness (§34.1): the definitions of $\\mathsf{NP}$-hard (a lower bound) and $\\mathsf{NP}$-complete (hard and in $\\mathsf{NP}$).\n[^clrs-cook]: **CLRS**, Ch. 34 — NP-Completeness (§34.3): the Cook–Levin theorem that **SAT** is $\\mathsf{NP}$-complete, proved by encoding a verifier's computation as a boolean formula.\n[^skiena-web]: **Skiena**, §11 — NP-Completeness: Karp's 1972 reductions and the web of $\\mathsf{NP}$-complete problems growing from satisfiability.\n[^erickson-gadget]: **Erickson**, Ch. 12 — NP-Hardness: the gadget reduction $\\textbf{3-SAT} \\le_P \\textbf{Independent-Set}$ (clause triangles plus conflict edges, $k = m$).\n[^clrs-recipe]: **CLRS**, Ch. 34 — NP-Completeness (§34.4): the standard four-step recipe for proving a new problem $\\mathsf{NP}$-complete.\n",{"text":8345,"minutes":8346,"time":8347,"words":8348},"7 min read",6.94,416400,1388,{"title":361,"description":8323},[8351,8353,8355],{"book":8032,"ref":8352},"Ch. 34 — NP-Completeness",{"book":8135,"ref":8354},"§11 — NP-Completeness",{"book":8166,"ref":8356},"Ch. 12 — 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