[{"data":1,"prerenderedAt":5907},["ShallowReactive",2],{"nav:algorithms":3,"lesson:\u002Falgorithms\u002Fintractability\u002Fcoping-with-hardness":374,"course-wordcounts":5775,"ref-card-index":5831},[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":368,"blurb":376,"body":377,"description":5737,"extension":5738,"meta":5739,"module":353,"navigation":5741,"path":369,"practice":5742,"rawbody":5757,"readingTime":5758,"seo":5763,"sources":5764,"status":5771,"stem":5772,"summary":373,"topics":5773,"__hash__":5774},"course\u002F01.algorithms\u002F12.intractability\u002F03.coping-with-hardness.md","",{"type":378,"value":379,"toc":5725},"minimark",[380,389,479,528,551,558,1069,1074,1089,1436,1580,2023,2028,2149,2204,2211,2280,2395,2539,2732,3197,3200,3392,3481,3792,3796,3814,3821,4041,4048,4068,4267,4278,4282,4298,4356,4436,4458,4714,4718,4748,5068,5071,5075,5078,5129,5133,5171,5178,5215,5219,5475,5721],[381,382,383,384,388],"p",{},"This is the capstone of the course, so let us first take stock of the toolkit we\nhave assembled. Almost every algorithm we built fell under a handful of\n",[385,386,387],"strong",{},"paradigms",", recurring shapes of attack that transcend any single problem:",[390,391,392,399,410,423,433,439],"ul",{},[393,394,395,398],"li",{},[385,396,397],{},"Divide and conquer."," Split, recurse, combine; analyzed by recurrences and\nthe master theorem (merge sort, counting inversions, closest points, Karatsuba,\nStrassen).",[393,400,401,404,405,409],{},[385,402,403],{},"Dynamic programming."," Tabulate overlapping subproblems (LCS, interval\nscheduling, Bellman–Ford, Floyd–Warshall, ",[406,407,408],"a",{"href":252},"knapsack",", subset-sum).",[393,411,412,417,418,422],{},[385,413,414,416],{},[406,415,218],{"href":211},"."," Commit to a locally best choice and prove it is safe (Huffman,\nDijkstra, Prim, Kruskal; note that Dijkstra and Prim share the ",[419,420,421],"em",{},"same"," greedy\nskeleton, differing only in the priority key).",[393,424,425,428,429,432],{},[385,426,427],{},"Graph methods."," BFS, DFS, topological sort, strongly connected components,\nand the shortest-path and ",[406,430,431],{"href":170},"spanning-tree"," algorithms built atop them.",[393,434,435,438],{},[385,436,437],{},"Network flow."," Max-flow \u002F min-cut (Ford–Fulkerson, Edmonds–Karp) as a\nmodeling lens that absorbs matching, scheduling, and connectivity problems.",[393,440,441,478],{},[385,442,443,446,447],{},[406,444,445],{"href":359},"Reductions"," and ",[406,448,449,477],{"href":364},[450,451,454],"span",{"className":452},[453],"katex",[450,455,459],{"className":456,"ariaHidden":458},[457],"katex-html","true",[450,460,463,468],{"className":461},[462],"base",[450,464],{"className":465,"style":467},[466],"strut","height:0.6944em;",[450,469,472],{"className":470},[471],"mord",[450,473,476],{"className":474},[471,475],"mathsf","NP","-completeness",", the meta-tool: relate one\nproblem to another, transferring either an algorithm or a hardness proof.",[381,480,481,482,500,501,519,520,523,524,527],{},"That last paradigm is the hinge on which this lesson turns. Suppose you have\nfollowed the recipe from the previous lesson and proved that the problem on your\ndesk is ",[450,483,485],{"className":484},[453],[450,486,488],{"className":487,"ariaHidden":458},[457],[450,489,491,494],{"className":490},[462],[450,492],{"className":493,"style":467},[466],[450,495,497],{"className":496},[471],[450,498,476],{"className":499},[471,475],"-hard. This is genuine progress: you now know not to waste\nmonths hunting for a fast exact algorithm that almost certainly does not exist.\nBut the problem has not gone away. The trucks still need routing, the chips still\nneed placing, the schedule still must be made. ",[450,502,504],{"className":503},[453],[450,505,507],{"className":506,"ariaHidden":458},[457],[450,508,510,513],{"className":509},[462],[450,511],{"className":512,"style":467},[466],[450,514,516],{"className":515},[471],[450,517,476],{"className":518},[471,475],"-hardness is a\nstatement about the ",[419,521,522],{},"worst case over all instances","; it does not forbid doing\nwell on the instances you actually meet, or doing ",[419,525,526],{},"nearly"," as well as optimal, or\ndoing exactly as well but slowly.",[381,529,530,531,534,535,538,539,542,543,546,547,550],{},"There are four honest ways to cope, and a well-designed system often blends\nthem. We can ",[385,532,533],{},"approximate",": settle for a solution provably close to optimal.\nWe can use ",[385,536,537],{},"heuristics",", methods that work well in practice without\nguarantees. We can pay for an ",[385,540,541],{},"exact exponential"," algorithm that is merely\n",[419,544,545],{},"smart"," about its exponential search. Or we can exploit ",[385,548,549],{},"special structure","\nin our instances. We take each in turn.",[381,552,553,554,557],{},"The decision tree below is the through-line of the whole subject in one picture:\n",[419,555,556],{},"first"," ask whether the problem is even hard; only the rightmost branch forces\nthe compromises of this lesson.",[559,560,564,1063],"figure",{"className":561},[562,563],"tikz-figure","tikz-diagram-rendered",[565,566,571],"svg",{"xmlns":567,"width":568,"height":569,"viewBox":570},"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg","417.230","515.701","-75 -75 312.923 386.776",[572,573,576,614,617,663,689,692,755,787,790,829,855,858,891,894,933,936,939,947,950,953,970,973,976,989,992,995,1001,1004,1007,1020,1023,1026,1032,1035,1038,1051,1054,1057],"g",{"stroke":574,"style":575},"currentColor","stroke-miterlimit:10;stroke-width:.4",[572,577,579,584],{"style":578},"stroke-width:1.2",[580,581],"path",{"fill":582,"d":583},"none","M94.572-71.67h-44.43a1 1 0 0 0-1 1v46.37a1 1 0 0 0 1 1h44.43a1 1 0 0 0 1-1v-46.37a1 1 0 0 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78.819)",[580,982],{"d":962,"fill":574,"stroke":574,"className":983,"style":946},[594],[572,985,986],{"transform":980},[580,987],{"d":968,"fill":574,"stroke":574,"className":988,"style":946},[594],[580,990],{"fill":582,"d":991},"M72.357 61.013V92.08",[580,993],{"d":994},"m72.357 94.586 1.35-3.584-1.35 1.178-1.351-1.178Z",[572,996,998],{"transform":997},"translate(3.533 127.076)",[580,999],{"d":944,"fill":574,"stroke":574,"className":1000,"style":946},[594],[580,1002],{"fill":582,"d":1003},"M45.903 119.94H3.453",[580,1005],{"d":1006},"m.948 119.94 3.584 1.351-1.178-1.35 1.178-1.351Z",[572,1008,1009,1015],{"stroke":582,"fontFamily":955,"fontSize":956},[572,1010,1012],{"transform":1011},"translate(-54.571 162.532)",[580,1013],{"d":962,"fill":574,"stroke":574,"className":1014,"style":946},[594],[572,1016,1017],{"transform":1011},[580,1018],{"d":968,"fill":574,"stroke":574,"className":1019,"style":946},[594],[580,1021],{"fill":582,"d":1022},"M72.357 144.726v31.067",[580,1024],{"d":1025},"m72.357 178.3 1.35-3.585-1.35 1.178-1.351-1.178Z",[572,1027,1029],{"transform":1028},"translate(3.533 210.79)",[580,1030],{"d":944,"fill":574,"stroke":574,"className":1031,"style":946},[594],[580,1033],{"fill":582,"d":1034},"M97.255 203.654h42.45",[580,1036],{"d":1037},"m142.21 203.654-3.585-1.351 1.18 1.35-1.18 1.351Z",[572,1039,1040,1046],{"stroke":582,"fontFamily":955,"fontSize":956},[572,1041,1043],{"transform":1042},"translate(42.306 246.245)",[580,1044],{"d":962,"fill":574,"stroke":574,"className":1045,"style":946},[594],[572,1047,1048],{"transform":1042},[580,1049],{"d":968,"fill":574,"stroke":574,"className":1050,"style":946},[594],[580,1052],{"fill":582,"d":1053},"M72.357 228.438v28.223",[580,1055],{"d":1056},"m72.357 259.167 1.35-3.585-1.35 1.18-1.351-1.18Z",[572,1058,1060],{"transform":1059},"translate(3.533 293.08)",[580,1061],{"d":944,"fill":574,"stroke":574,"className":1062,"style":946},[594],[1064,1065,1068],"figcaption",{"className":1066},[1067],"tikz-cap","Decision tree choosing a coping strategy for a possibly NP-hard problem.",[1070,1071,1073],"h2",{"id":1072},"approximation-algorithms","Approximation algorithms",[381,1075,1076,1077,1080,1081,1084,1085,1088],{},"The most intellectually satisfying response is to relax ",[419,1078,1079],{},"optimality"," while\nkeeping a ",[419,1082,1083],{},"guarantee",". An ",[385,1086,1087],{},"approximation algorithm"," runs in polynomial time\nand returns a solution provably within a bounded factor of the best possible.",[1090,1091,1093],"callout",{"type":1092},"definition",[381,1094,1095,1117,1118,1133,1134,1174,1175,1193,1194,1299,1300,1344,1345,1360,1361,1364,1365,1435],{},[385,1096,1097,1098,1116],{},"Definition (",[450,1099,1101],{"className":1100},[453],[450,1102,1104],{"className":1103,"ariaHidden":458},[457],[450,1105,1107,1111],{"className":1106},[462],[450,1108],{"className":1109,"style":1110},[466],"height:0.625em;vertical-align:-0.1944em;",[450,1112,1115],{"className":1113},[471,1114],"mathnormal","ρ","-approximation)."," For a minimization problem, an algorithm is a ",[450,1119,1121],{"className":1120},[453],[450,1122,1124],{"className":1123,"ariaHidden":458},[457],[450,1125,1127,1130],{"className":1126},[462],[450,1128],{"className":1129,"style":1110},[466],[450,1131,1115],{"className":1132},[471,1114],"-approximation (with\n",[450,1135,1137],{"className":1136},[453],[450,1138,1140,1163],{"className":1139,"ariaHidden":458},[457],[450,1141,1143,1147,1150,1155,1160],{"className":1142},[462],[450,1144],{"className":1145,"style":1146},[466],"height:0.8304em;vertical-align:-0.1944em;",[450,1148,1115],{"className":1149},[471,1114],[450,1151],{"className":1152,"style":1154},[1153],"mspace","margin-right:0.2778em;",[450,1156,1159],{"className":1157},[1158],"mrel","≥",[450,1161],{"className":1162,"style":1154},[1153],[450,1164,1166,1170],{"className":1165},[462],[450,1167],{"className":1168,"style":1169},[466],"height:0.6444em;",[450,1171,1173],{"className":1172},[471],"1",") if, on every instance, it runs in polynomial time and returns a\nsolution of cost ",[450,1176,1178],{"className":1177},[453],[450,1179,1181],{"className":1180,"ariaHidden":458},[457],[450,1182,1184,1188],{"className":1183},[462],[450,1185],{"className":1186,"style":1187},[466],"height:0.6833em;",[450,1189,1192],{"className":1190,"style":1191},[471,1114],"margin-right:0.0715em;","C"," satisfying\n",[450,1195,1197],{"className":1196},[453],[450,1198,1200,1220,1242],{"className":1199,"ariaHidden":458},[457],[450,1201,1203,1207,1210,1213,1217],{"className":1202},[462],[450,1204],{"className":1205,"style":1206},[466],"height:0.8193em;vertical-align:-0.136em;",[450,1208,1192],{"className":1209,"style":1191},[471,1114],[450,1211],{"className":1212,"style":1154},[1153],[450,1214,1216],{"className":1215},[1158],"≤",[450,1218],{"className":1219,"style":1154},[1153],[450,1221,1223,1227,1230,1234,1239],{"className":1222},[462],[450,1224],{"className":1225,"style":1226},[466],"height:0.6389em;vertical-align:-0.1944em;",[450,1228,1115],{"className":1229},[471,1114],[450,1231],{"className":1232,"style":1233},[1153],"margin-right:0.2222em;",[450,1235,1238],{"className":1236},[1237],"mbin","⋅",[450,1240],{"className":1241,"style":1233},[1153],[450,1243,1245,1249,1294],{"className":1244},[462],[450,1246],{"className":1247,"style":1248},[466],"height:0.8831em;vertical-align:-0.1944em;",[450,1250,1252,1255],{"className":1251},[471],[450,1253,1192],{"className":1254,"style":1191},[471,1114],[450,1256,1259],{"className":1257},[1258],"msupsub",[450,1260,1263],{"className":1261},[1262],"vlist-t",[450,1264,1267],{"className":1265},[1266],"vlist-r",[450,1268,1272],{"className":1269,"style":1271},[1270],"vlist","height:0.6887em;",[450,1273,1275,1280],{"style":1274},"top:-3.063em;margin-right:0.05em;",[450,1276],{"className":1277,"style":1279},[1278],"pstrut","height:2.7em;",[450,1281,1287],{"className":1282},[1283,1284,1285,1286],"sizing","reset-size6","size3","mtight",[450,1288,1290],{"className":1289},[471,1286],[450,1291,1293],{"className":1292},[471,1286],"∗",[450,1295,1298],{"className":1296},[1297],"mpunct",",","\nwhere ",[450,1301,1303],{"className":1302},[453],[450,1304,1306],{"className":1305,"ariaHidden":458},[457],[450,1307,1309,1312],{"className":1308},[462],[450,1310],{"className":1311,"style":1271},[466],[450,1313,1315,1318],{"className":1314},[471],[450,1316,1192],{"className":1317,"style":1191},[471,1114],[450,1319,1321],{"className":1320},[1258],[450,1322,1324],{"className":1323},[1262],[450,1325,1327],{"className":1326},[1266],[450,1328,1330],{"className":1329,"style":1271},[1270],[450,1331,1332,1335],{"style":1274},[450,1333],{"className":1334,"style":1279},[1278],[450,1336,1338],{"className":1337},[1283,1284,1285,1286],[450,1339,1341],{"className":1340},[471,1286],[450,1342,1293],{"className":1343},[471,1286]," is the cost of an optimal solution. The factor ",[450,1346,1348],{"className":1347},[453],[450,1349,1351],{"className":1350,"ariaHidden":458},[457],[450,1352,1354,1357],{"className":1353},[462],[450,1355],{"className":1356,"style":1110},[466],[450,1358,1115],{"className":1359},[471,1114]," is the\n",[385,1362,1363],{},"approximation ratio",". (For a maximization problem the bound flips to ",[450,1366,1368],{"className":1367},[453],[450,1369,1371,1389],{"className":1370,"ariaHidden":458},[457],[450,1372,1374,1377,1380,1383,1386],{"className":1373},[462],[450,1375],{"className":1376,"style":1206},[466],[450,1378,1192],{"className":1379,"style":1191},[471,1114],[450,1381],{"className":1382,"style":1154},[1153],[450,1384,1159],{"className":1385},[1158],[450,1387],{"className":1388,"style":1154},[1153],[450,1390,1392,1396,1428,1432],{"className":1391},[462],[450,1393],{"className":1394,"style":1395},[466],"height:1em;vertical-align:-0.25em;",[450,1397,1399,1402],{"className":1398},[471],[450,1400,1192],{"className":1401,"style":1191},[471,1114],[450,1403,1405],{"className":1404},[1258],[450,1406,1408],{"className":1407},[1262],[450,1409,1411],{"className":1410},[1266],[450,1412,1414],{"className":1413,"style":1271},[1270],[450,1415,1416,1419],{"style":1274},[450,1417],{"className":1418,"style":1279},[1278],[450,1420,1422],{"className":1421},[1283,1284,1285,1286],[450,1423,1425],{"className":1424},[471,1286],[450,1426,1293],{"className":1427},[471,1286],[450,1429,1431],{"className":1430},[471],"\u002F",[450,1433,1115],{"className":1434},[471,1114],".)",[381,1437,1438,1439,1474,1475,1479,1480,1527,1528,1531,1532,1576,1577,416],{},"A ratio of ",[450,1440,1442],{"className":1441},[453],[450,1443,1445,1464],{"className":1444,"ariaHidden":458},[457],[450,1446,1448,1451,1454,1457,1461],{"className":1447},[462],[450,1449],{"className":1450,"style":1110},[466],[450,1452,1115],{"className":1453},[471,1114],[450,1455],{"className":1456,"style":1154},[1153],[450,1458,1460],{"className":1459},[1158],"=",[450,1462],{"className":1463,"style":1154},[1153],[450,1465,1467,1470],{"className":1466},[462],[450,1468],{"className":1469,"style":1169},[466],[450,1471,1473],{"className":1472},[471],"2"," means ",[1476,1477,1478],"q",{},"never more than twice optimal",": guaranteed, on\nevery input, with no exceptions. The subtlety, and the beauty, is that we prove\nthis ",[419,1481,1482,1483],{},"without ever knowing ",[450,1484,1486],{"className":1485},[453],[450,1487,1489],{"className":1488,"ariaHidden":458},[457],[450,1490,1492,1495],{"className":1491},[462],[450,1493],{"className":1494,"style":1271},[466],[450,1496,1498,1501],{"className":1497},[471],[450,1499,1192],{"className":1500,"style":1191},[471,1114],[450,1502,1504],{"className":1503},[1258],[450,1505,1507],{"className":1506},[1262],[450,1508,1510],{"className":1509},[1266],[450,1511,1513],{"className":1512,"style":1271},[1270],[450,1514,1515,1518],{"style":1274},[450,1516],{"className":1517,"style":1279},[1278],[450,1519,1521],{"className":1520},[1283,1284,1285,1286],[450,1522,1524],{"className":1523},[471,1286],[450,1525,1293],{"className":1526},[471,1286],". The trick is almost always a ",[385,1529,1530],{},"lower\nbound",": find some quantity that provably under-estimates ",[450,1533,1535],{"className":1534},[453],[450,1536,1538],{"className":1537,"ariaHidden":458},[457],[450,1539,1541,1544],{"className":1540},[462],[450,1542],{"className":1543,"style":1271},[466],[450,1545,1547,1550],{"className":1546},[471],[450,1548,1192],{"className":1549,"style":1191},[471,1114],[450,1551,1553],{"className":1552},[1258],[450,1554,1556],{"className":1555},[1262],[450,1557,1559],{"className":1558},[1266],[450,1560,1562],{"className":1561,"style":1271},[1270],[450,1563,1564,1567],{"style":1274},[450,1565],{"className":1566,"style":1279},[1278],[450,1568,1570],{"className":1569},[1283,1284,1285,1286],[450,1571,1573],{"className":1572},[471,1286],[450,1574,1293],{"className":1575},[471,1286],", then show\nyour solution is not much bigger than 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",[450,1845,1847],{"className":1846},[453],[450,1848,1850],{"className":1849,"ariaHidden":458},[457],[450,1851,1853,1856],{"className":1852},[462],[450,1854],{"className":1855,"style":1187},[466],[450,1857,1192],{"className":1858,"style":1191},[471,1114]," into the band ",[450,1861,1863],{"className":1862},[453],[450,1864,1866],{"className":1865,"ariaHidden":458},[457],[450,1867,1869,1872,1877,1909,1912,1916,1919,1922,1954],{"className":1868},[462],[450,1870],{"className":1871,"style":1395},[466],[450,1873,1876],{"className":1874},[1875],"mopen","[",[450,1878,1880,1883],{"className":1879},[471],[450,1881,1192],{"className":1882,"style":1191},[471,1114],[450,1884,1886],{"className":1885},[1258],[450,1887,1889],{"className":1888},[1262],[450,1890,1892],{"className":1891},[1266],[450,1893,1895],{"className":1894,"style":1271},[1270],[450,1896,1897,1900],{"style":1274},[450,1898],{"className":1899,"style":1279},[1278],[450,1901,1903],{"className":1902},[1283,1284,1285,1286],[450,1904,1906],{"className":1905},[471,1286],[450,1907,1293],{"className":1908},[471,1286],[450,1910,1298],{"className":1911},[1297],[450,1913],{"className":1914,"style":1915},[1153],"margin-right:0.1667em;",[450,1917,1115],{"className":1918},[471,1114],[450,1920],{"className":1921,"style":1915},[1153],[450,1923,1925,1928],{"className":1924},[471],[450,1926,1192],{"className":1927,"style":1191},[471,1114],[450,1929,1931],{"className":1930},[1258],[450,1932,1934],{"className":1933},[1262],[450,1935,1937],{"className":1936},[1266],[450,1938,1940],{"className":1939,"style":1271},[1270],[450,1941,1942,1945],{"style":1274},[450,1943],{"className":1944,"style":1279},[1278],[450,1946,1948],{"className":1947},[1283,1284,1285,1286],[450,1949,1951],{"className":1950},[471,1286],[450,1952,1293],{"className":1953},[471,1286],[450,1955,1958],{"className":1956},[1957],"mclose","]"," — proved via a lower bound ",[450,1961,1963],{"className":1962},[453],[450,1964,1966,1985],{"className":1965,"ariaHidden":458},[457],[450,1967,1969,1972,1976,1979,1982],{"className":1968},[462],[450,1970],{"className":1971,"style":1206},[466],[450,1973,1975],{"className":1974},[471,1114],"L",[450,1977],{"className":1978,"style":1154},[1153],[450,1980,1216],{"className":1981},[1158],[450,1983],{"className":1984,"style":1154},[1153],[450,1986,1988,1991],{"className":1987},[462],[450,1989],{"className":1990,"style":1271},[466],[450,1992,1994,1997],{"className":1993},[471],[450,1995,1192],{"className":1996,"style":1191},[471,1114],[450,1998,2000],{"className":1999},[1258],[450,2001,2003],{"className":2002},[1262],[450,2004,2006],{"className":2005},[1266],[450,2007,2009],{"className":2008,"style":1271},[1270],[450,2010,2011,2014],{"style":1274},[450,2012],{"className":2013,"style":1279},[1278],[450,2015,2017],{"className":2016},[1283,1284,1285,1286],[450,2018,2020],{"className":2019},[471,1286],[450,2021,1293],{"className":2022},[471,1286],[2024,2025,2027],"h3",{"id":2026},"a-worked-example-2-approximation-for-vertex-cover","A worked example: 2-approximation for Vertex Cover",[381,2029,1827,2030,2033,2034,2088,2089,2114,2115,2133,2134,416],{},[385,2031,2032],{},"vertex cover"," of a graph ",[450,2035,2037],{"className":2036},[453],[450,2038,2040,2059],{"className":2039,"ariaHidden":458},[457],[450,2041,2043,2046,2050,2053,2056],{"className":2042},[462],[450,2044],{"className":2045,"style":1187},[466],[450,2047,2049],{"className":2048},[471,1114],"G",[450,2051],{"className":2052,"style":1154},[1153],[450,2054,1460],{"className":2055},[1158],[450,2057],{"className":2058,"style":1154},[1153],[450,2060,2062,2065,2069,2073,2076,2079,2084],{"className":2061},[462],[450,2063],{"className":2064,"style":1395},[466],[450,2066,2068],{"className":2067},[1875],"(",[450,2070,2072],{"className":2071,"style":1233},[471,1114],"V",[450,2074,1298],{"className":2075},[1297],[450,2077],{"className":2078,"style":1915},[1153],[450,2080,2083],{"className":2081,"style":2082},[471,1114],"margin-right:0.0576em;","E",[450,2085,2087],{"className":2086},[1957],")"," is a set of vertices touching every\nedge: for each edge, at least one endpoint is chosen. ",[450,2090,2092],{"className":2091},[453],[450,2093,2095],{"className":2094,"ariaHidden":458},[457],[450,2096,2098,2101],{"className":2097},[462],[450,2099],{"className":2100,"style":1187},[466],[450,2102,2106],{"className":2103},[2104,2105],"enclosing","textsc",[450,2107,2110],{"className":2108},[471,2109],"text",[450,2111,2113],{"className":2112},[471],"Min-Vertex-Cover",", the\nproblem of finding the smallest such set, is ",[450,2116,2118],{"className":2117},[453],[450,2119,2121],{"className":2120,"ariaHidden":458},[457],[450,2122,2124,2127],{"className":2123},[462],[450,2125],{"className":2126,"style":467},[466],[450,2128,2130],{"className":2129},[471],[450,2131,476],{"className":2132},[471,475],"-hard. Yet a\ndisarmingly simple algorithm comes within a factor of ",[450,2135,2137],{"className":2136},[453],[450,2138,2140],{"className":2139,"ariaHidden":458},[457],[450,2141,2143,2146],{"className":2142},[462],[450,2144],{"className":2145,"style":1169},[466],[450,2147,1473],{"className":2148},[471],[2150,2151,2155],"pre",{"className":2152,"code":2153,"language":2154,"meta":376,"style":376},"language-algorithm shiki shiki-themes Vesper Light - Orange Boost (Quick Open Adjusted) vesper","caption: $\\textsc{Approx-Vertex-Cover}(G)$ — return a cover at most twice optimal\nnumber: 1\n$C \\gets \\emptyset$\n$E' \\gets E$ \u002F\u002F working copy\nwhile $E' \\neq \\emptyset$ do\n  pick any edge $(u, v) \\in E'$\n  $C \\gets C \\cup \\{u, v\\}$ \u002F\u002F take both endpoints\n  remove from $E'$ every edge incident to $u$ or $v$\nreturn $C$\n","algorithm",[2156,2157,2158,2164,2169,2174,2179,2184,2189,2194,2199],"code",{"__ignoreMap":376},[450,2159,2161],{"class":2160,"line":6},"line",[450,2162,2163],{},"caption: $\\textsc{Approx-Vertex-Cover}(G)$ — return a cover at most twice optimal\n",[450,2165,2166],{"class":2160,"line":18},[450,2167,2168],{},"number: 1\n",[450,2170,2171],{"class":2160,"line":24},[450,2172,2173],{},"$C \\gets \\emptyset$\n",[450,2175,2176],{"class":2160,"line":73},[450,2177,2178],{},"$E' \\gets E$ \u002F\u002F working copy\n",[450,2180,2181],{"class":2160,"line":102},[450,2182,2183],{},"while $E' \\neq \\emptyset$ do\n",[450,2185,2186],{"class":2160,"line":108},[450,2187,2188],{},"  pick any edge $(u, v) \\in E'$\n",[450,2190,2191],{"class":2160,"line":116},[450,2192,2193],{},"  $C \\gets C \\cup \\{u, v\\}$ \u002F\u002F take both endpoints\n",[450,2195,2196],{"class":2160,"line":196},[450,2197,2198],{},"  remove from $E'$ every edge incident to $u$ or $v$\n",[450,2200,2201],{"class":2160,"line":202},[450,2202,2203],{},"return $C$\n",[381,2205,2206,2207,2210],{},"The algorithm repeatedly grabs an uncovered edge and throws ",[419,2208,2209],{},"both"," its\nendpoints into the cover. Taking both feels wasteful (surely one would do?),\nbut it is exactly what makes the analysis work.",[381,2212,2213,2216,2217,2263,2264,2279],{},[385,2214,2215],{},"Correctness."," The loop continues until ",[450,2218,2220],{"className":2219},[453],[450,2221,2223],{"className":2222,"ariaHidden":458},[457],[450,2224,2226,2230],{"className":2225},[462],[450,2227],{"className":2228,"style":2229},[466],"height:0.7519em;",[450,2231,2233,2236],{"className":2232},[471],[450,2234,2083],{"className":2235,"style":2082},[471,1114],[450,2237,2239],{"className":2238},[1258],[450,2240,2242],{"className":2241},[1262],[450,2243,2245],{"className":2244},[1266],[450,2246,2248],{"className":2247,"style":2229},[1270],[450,2249,2250,2253],{"style":1274},[450,2251],{"className":2252,"style":1279},[1278],[450,2254,2256],{"className":2255},[1283,1284,1285,1286],[450,2257,2259],{"className":2258},[471,1286],[450,2260,2262],{"className":2261},[471,1286],"′"," is empty, i.e. until every edge\nis incident to some chosen vertex. So ",[450,2265,2267],{"className":2266},[453],[450,2268,2270],{"className":2269,"ariaHidden":458},[457],[450,2271,2273,2276],{"className":2272},[462],[450,2274],{"className":2275,"style":1187},[466],[450,2277,1192],{"className":2278,"style":1191},[471,1114]," is a valid vertex cover.",[381,2281,2282,2285,2286,2303,2304,2307,2308,2323,2324,2357,2358,2374,2375,2390,2391,2394],{},[385,2283,2284],{},"The ratio."," Let ",[450,2287,2289],{"className":2288},[453],[450,2290,2292],{"className":2291,"ariaHidden":458},[457],[450,2293,2295,2298],{"className":2294},[462],[450,2296],{"className":2297,"style":1187},[466],[450,2299,2302],{"className":2300,"style":2301},[471,1114],"margin-right:0.109em;","M"," be the set of edges ",[419,2305,2306],{},"picked"," in the loop (one per\niteration). No two edges in ",[450,2309,2311],{"className":2310},[453],[450,2312,2314],{"className":2313,"ariaHidden":458},[457],[450,2315,2317,2320],{"className":2316},[462],[450,2318],{"className":2319,"style":1187},[466],[450,2321,2302],{"className":2322,"style":2301},[471,1114]," share an endpoint: once we pick ",[450,2325,2327],{"className":2326},[453],[450,2328,2330],{"className":2329,"ariaHidden":458},[457],[450,2331,2333,2336,2339,2343,2346,2349,2354],{"className":2332},[462],[450,2334],{"className":2335,"style":1395},[466],[450,2337,2068],{"className":2338},[1875],[450,2340,2342],{"className":2341},[471,1114],"u",[450,2344,1298],{"className":2345},[1297],[450,2347],{"className":2348,"style":1915},[1153],[450,2350,2353],{"className":2351,"style":2352},[471,1114],"margin-right:0.0359em;","v",[450,2355,2087],{"className":2356},[1957],", we\ndelete every edge touching ",[450,2359,2361],{"className":2360},[453],[450,2362,2364],{"className":2363,"ariaHidden":458},[457],[450,2365,2367,2371],{"className":2366},[462],[450,2368],{"className":2369,"style":2370},[466],"height:0.4306em;",[450,2372,2342],{"className":2373},[471,1114]," or ",[450,2376,2378],{"className":2377},[453],[450,2379,2381],{"className":2380,"ariaHidden":458},[457],[450,2382,2384,2387],{"className":2383},[462],[450,2385],{"className":2386,"style":2370},[466],[450,2388,2353],{"className":2389,"style":2352},[471,1114],", so no later picked edge can reuse them.\nA set of edges sharing no endpoints is a ",[385,2392,2393],{},"matching",". Now the two key facts:",[390,2396,2397,2448],{},[393,2398,2399,2400,2447],{},"Our cover has ",[450,2401,2403],{"className":2402},[453],[450,2404,2406,2431],{"className":2405,"ariaHidden":458},[457],[450,2407,2409,2412,2416,2419,2422,2425,2428],{"className":2408},[462],[450,2410],{"className":2411,"style":1395},[466],[450,2413,2415],{"className":2414},[471],"∣",[450,2417,1192],{"className":2418,"style":1191},[471,1114],[450,2420,2415],{"className":2421},[471],[450,2423],{"className":2424,"style":1154},[1153],[450,2426,1460],{"className":2427},[1158],[450,2429],{"className":2430,"style":1154},[1153],[450,2432,2434,2437,2441,2444],{"className":2433},[462],[450,2435],{"className":2436,"style":1395},[466],[450,2438,2440],{"className":2439},[471],"2∣",[450,2442,2302],{"className":2443,"style":2301},[471,1114],[450,2445,2415],{"className":2446},[471],", two fresh vertices per picked edge.",[393,2449,2450,2453,2454,2469,2470,416],{},[419,2451,2452],{},"Any"," vertex cover, including the optimal one, must contain at least one\nendpoint of each edge in ",[450,2455,2457],{"className":2456},[453],[450,2458,2460],{"className":2459,"ariaHidden":458},[457],[450,2461,2463,2466],{"className":2462},[462],[450,2464],{"className":2465,"style":1187},[466],[450,2467,2302],{"className":2468,"style":2301},[471,1114],"; since those endpoints are all distinct, ",[450,2471,2473],{"className":2472},[453],[450,2474,2476,2524],{"className":2475,"ariaHidden":458},[457],[450,2477,2479,2483,2515,2518,2521],{"className":2478},[462],[450,2480],{"className":2481,"style":2482},[466],"height:0.8247em;vertical-align:-0.136em;",[450,2484,2486,2489],{"className":2485},[471],[450,2487,1192],{"className":2488,"style":1191},[471,1114],[450,2490,2492],{"className":2491},[1258],[450,2493,2495],{"className":2494},[1262],[450,2496,2498],{"className":2497},[1266],[450,2499,2501],{"className":2500,"style":1271},[1270],[450,2502,2503,2506],{"style":1274},[450,2504],{"className":2505,"style":1279},[1278],[450,2507,2509],{"className":2508},[1283,1284,1285,1286],[450,2510,2512],{"className":2511},[471,1286],[450,2513,1293],{"className":2514},[471,1286],[450,2516],{"className":2517,"style":1154},[1153],[450,2519,1159],{"className":2520},[1158],[450,2522],{"className":2523,"style":1154},[1153],[450,2525,2527,2530,2533,2536],{"className":2526},[462],[450,2528],{"className":2529,"style":1395},[466],[450,2531,2415],{"className":2532},[471],[450,2534,2302],{"className":2535,"style":2301},[471,1114],[450,2537,2415],{"className":2538},[471],[381,2540,2541,2542,2643,2644,2667,2668,2683,2684,2693,2694,2715,2716,2731],{},"Chaining these,\n",[450,2543,2545],{"className":2544},[453],[450,2546,2548,2572,2596],{"className":2547,"ariaHidden":458},[457],[450,2549,2551,2554,2557,2560,2563,2566,2569],{"className":2550},[462],[450,2552],{"className":2553,"style":1395},[466],[450,2555,2415],{"className":2556},[471],[450,2558,1192],{"className":2559,"style":1191},[471,1114],[450,2561,2415],{"className":2562},[471],[450,2564],{"className":2565,"style":1154},[1153],[450,2567,1460],{"className":2568},[1158],[450,2570],{"className":2571,"style":1154},[1153],[450,2573,2575,2578,2581,2584,2587,2590,2593],{"className":2574},[462],[450,2576],{"className":2577,"style":1395},[466],[450,2579,2440],{"className":2580},[471],[450,2582,2302],{"className":2583,"style":2301},[471,1114],[450,2585,2415],{"className":2586},[471],[450,2588],{"className":2589,"style":1154},[1153],[450,2591,1216],{"className":2592},[1158],[450,2594],{"className":2595,"style":1154},[1153],[450,2597,2599,2602,2605,2608,2640],{"className":2598},[462],[450,2600],{"className":2601,"style":1248},[466],[450,2603,1473],{"className":2604},[471],[450,2606],{"className":2607,"style":1915},[1153],[450,2609,2611,2614],{"className":2610},[471],[450,2612,1192],{"className":2613,"style":1191},[471,1114],[450,2615,2617],{"className":2616},[1258],[450,2618,2620],{"className":2619},[1262],[450,2621,2623],{"className":2622},[1266],[450,2624,2626],{"className":2625,"style":1271},[1270],[450,2627,2628,2631],{"style":1274},[450,2629],{"className":2630,"style":1279},[1278],[450,2632,2634],{"className":2633},[1283,1284,1285,1286],[450,2635,2637],{"className":2636},[471,1286],[450,2638,1293],{"className":2639},[471,1286],[450,2641,1298],{"className":2642},[1297],"\nso ",[450,2645,2647],{"className":2646},[453],[450,2648,2650],{"className":2649,"ariaHidden":458},[457],[450,2651,2653,2657],{"className":2652},[462],[450,2654],{"className":2655,"style":2656},[466],"height:0.8778em;vertical-align:-0.1944em;",[450,2658,2660],{"className":2659},[2104,2105],[450,2661,2663],{"className":2662},[471,2109],[450,2664,2666],{"className":2665},[471],"Approx-Vertex-Cover"," is a ",[450,2669,2671],{"className":2670},[453],[450,2672,2674],{"className":2673,"ariaHidden":458},[457],[450,2675,2677,2680],{"className":2676},[462],[450,2678],{"className":2679,"style":1169},[466],[450,2681,1473],{"className":2682},[471],"-approximation.",[2685,2686,2687],"sup",{},[406,2688,1173],{"href":2689,"ariaDescribedBy":2690,"dataFootnoteRef":376,"id":2692},"#user-content-fn-clrs-vc",[2691],"footnote-label","user-content-fnref-clrs-vc"," We bounded our output\nagainst ",[450,2695,2697],{"className":2696},[453],[450,2698,2700],{"className":2699,"ariaHidden":458},[457],[450,2701,2703,2706,2709,2712],{"className":2702},[462],[450,2704],{"className":2705,"style":1395},[466],[450,2707,2415],{"className":2708},[471],[450,2710,2302],{"className":2711,"style":2301},[471,1114],[450,2713,2415],{"className":2714},[471],", a quantity that lower-bounds the unknown optimum, the\nsignature move of approximation analysis. 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204.652-19.465L204.652-20.863Q204.652-21.133 204.544-21.195Q204.436-21.256 204.125-21.256L204.125-21.537L205.202-21.612L205.202-17.428Q205.202-17.240 205.257-17.157Q205.311-17.073 205.412-17.054Q205.513-17.035 205.728-17.035L205.728-16.755L204.621-16.687L204.621-17.104Q204.204-16.687 203.578-16.687Q203.148-16.687 202.775-16.899Q202.403-17.110 202.182-17.471Q201.962-17.832 201.962-18.266M203.636-16.909Q203.845-16.909 204.031-16.981Q204.218-17.052 204.371-17.189Q204.525-17.326 204.621-17.504L204.621-19.113Q204.535-19.260 204.390-19.380Q204.245-19.500 204.076-19.559Q203.906-19.619 203.725-19.619Q203.165-19.619 202.896-19.230Q202.628-18.840 202.628-18.259Q202.628-17.688 202.862-17.298Q203.096-16.909 203.636-16.909",[594],[572,3066,3067],{"transform":2892},[580,3068],{"d":3069,"fill":574,"stroke":574,"className":3070,"style":946},"M210.679-16.782L209.551-19.281Q209.479-19.428 209.349-19.460Q209.219-19.493 208.990-19.493L208.990-19.773L210.504-19.773L210.504-19.493Q210.152-19.493 210.152-19.346Q210.152-19.301 210.163-19.281L211.027-17.363L211.807-19.093Q211.841-19.161 211.841-19.240Q211.841-19.353 211.757-19.423Q211.673-19.493 211.554-19.493L211.554-19.773L212.750-19.773L212.750-19.493Q212.531-19.493 212.360-19.390Q212.190-19.288 212.101-19.093L211.065-16.782Q211.017-16.687 210.911-16.687L210.833-16.687Q210.727-16.687 210.679-16.782",[594],[572,3072,3073],{"transform":2892},[580,3074],{"d":3075,"fill":574,"stroke":574,"className":3076,"style":946},"M213.034-18.290Q213.034-18.611 213.159-18.900Q213.284-19.189 213.510-19.412Q213.735-19.636 214.031-19.756Q214.326-19.876 214.644-19.876Q214.972-19.876 215.234-19.776Q215.495-19.677 215.671-19.495Q215.847-19.312 215.941-19.054Q216.035-18.796 216.035-18.464Q216.035-18.372 215.953-18.351L213.698-18.351L213.698-18.290Q213.698-17.702 213.981-17.319Q214.265-16.936 214.832-16.936Q215.154-16.936 215.422-17.129Q215.690-17.322 215.779-17.637Q215.786-17.678 215.861-17.692L215.953-17.692Q216.035-17.668 216.035-17.596Q216.035-17.589 216.029-17.562Q215.916-17.165 215.545-16.926Q215.174-16.687 214.750-16.687Q214.313-16.687 213.913-16.895Q213.513-17.104 213.274-17.471Q213.034-17.838 213.034-18.290M213.704-18.560L215.519-18.560Q215.519-18.837 215.422-19.089Q215.324-19.342 215.126-19.498Q214.928-19.653 214.644-19.653Q214.367-19.653 214.154-19.495Q213.940-19.336 213.822-19.081Q213.704-18.826 213.704-18.560M218.373-16.755L216.637-16.755L216.637-17.035Q216.866-17.035 217.015-17.069Q217.163-17.104 217.163-17.244L217.163-19.093Q217.163-19.363 217.056-19.424Q216.948-19.486 216.637-19.486L216.637-19.766L217.666-19.841L217.666-19.134Q217.796-19.442 218.038-19.641Q218.281-19.841 218.599-19.841Q218.818-19.841 218.989-19.717Q219.159-19.592 219.159-19.380Q219.159-19.243 219.060-19.144Q218.961-19.045 218.828-19.045Q218.691-19.045 218.592-19.144Q218.493-19.243 218.493-19.380Q218.493-19.520 218.592-19.619Q218.302-19.619 218.102-19.423Q217.902-19.226 217.809-18.932Q217.717-18.638 217.717-18.358L217.717-17.244Q217.717-17.035 218.373-17.035L218.373-16.755M220.270-17.596L220.270-19.493L219.631-19.493L219.631-19.715Q219.949-19.715 220.166-19.925Q220.383-20.135 220.484-20.445Q220.585-20.754 220.585-21.062L220.851-21.062L220.851-19.773L221.928-19.773L221.928-19.493L220.851-19.493L220.851-17.609Q220.851-17.333 220.956-17.134Q221.060-16.936 221.320-16.936Q221.477-16.936 221.583-17.040Q221.689-17.145 221.738-17.298Q221.788-17.452 221.788-17.609L221.788-18.023L222.054-18.023L222.054-17.596Q222.054-17.370 221.955-17.160Q221.856-16.950 221.672-16.818Q221.487-16.687 221.258-16.687Q220.821-16.687 220.545-16.924Q220.270-17.162 220.270-17.596M222.823-18.290Q222.823-18.611 222.948-18.900Q223.073-19.189 223.299-19.412Q223.524-19.636 223.820-19.756Q224.115-19.876 224.433-19.876Q224.761-19.876 225.023-19.776Q225.284-19.677 225.460-19.495Q225.636-19.312 225.730-19.054Q225.824-18.796 225.824-18.464Q225.824-18.372 225.742-18.351L223.487-18.351L223.487-18.290Q223.487-17.702 223.770-17.319Q224.054-16.936 224.621-16.936Q224.943-16.936 225.211-17.129Q225.479-17.322 225.568-17.637Q225.575-17.678 225.650-17.692L225.742-17.692Q225.824-17.668 225.824-17.596Q225.824-17.589 225.818-17.562Q225.705-17.165 225.334-16.926Q224.963-16.687 224.539-16.687Q224.102-16.687 223.702-16.895Q223.302-17.104 223.063-17.471Q222.823-17.838 222.823-18.290M223.493-18.560L225.308-18.560Q225.308-18.837 225.211-19.089Q225.114-19.342 224.915-19.498Q224.717-19.653 224.433-19.653Q224.157-19.653 223.943-19.495Q223.729-19.336 223.611-19.081Q223.493-18.826 223.493-18.560M227.595-16.755L226.272-16.755L226.272-17.035Q226.833-17.035 227.212-17.435L227.927-18.232L227.014-19.281Q226.877-19.428 226.729-19.460Q226.580-19.493 226.313-19.493L226.313-19.773L227.814-19.773L227.814-19.493Q227.622-19.493 227.622-19.359Q227.622-19.329 227.653-19.281L228.248-18.597L228.689-19.093Q228.802-19.223 228.802-19.339Q228.802-19.401 228.764-19.447Q228.726-19.493 228.668-19.493L228.668-19.773L229.984-19.773L229.984-19.493Q229.424-19.493 229.044-19.093L228.422-18.392L229.417-17.244Q229.516-17.145 229.617-17.100Q229.718-17.056 229.829-17.046Q229.940-17.035 230.117-17.035L230.117-16.755L228.624-16.755L228.624-17.035Q228.689-17.035 228.749-17.069Q228.808-17.104 228.808-17.169Q228.808-17.216 228.778-17.244L228.101-18.030L227.568-17.435Q227.455-17.305 227.455-17.189Q227.455-17.124 227.496-17.080Q227.537-17.035 227.595-17.035",[594],[1064,3078,3080,3101,3102,416],{"className":3079},[1067],[450,3081,3083],{"className":3082},[453],[450,3084,3086],{"className":3085,"ariaHidden":458},[457],[450,3087,3089,3092],{"className":3088},[462],[450,3090],{"className":3091,"style":2656},[466],[450,3093,3095],{"className":3094},[2104,2105],[450,3096,3098],{"className":3097},[471,2109],[450,3099,2666],{"className":3100},[471],": each picked matching edge (heavy) contributes both endpoints (filled) to the cover, giving ",[450,3103,3105],{"className":3104},[453],[450,3106,3108,3132,3156],{"className":3107,"ariaHidden":458},[457],[450,3109,3111,3114,3117,3120,3123,3126,3129],{"className":3110},[462],[450,3112],{"className":3113,"style":1395},[466],[450,3115,2415],{"className":3116},[471],[450,3118,1192],{"className":3119,"style":1191},[471,1114],[450,3121,2415],{"className":3122},[471],[450,3124],{"className":3125,"style":1154},[1153],[450,3127,1460],{"className":3128},[1158],[450,3130],{"className":3131,"style":1154},[1153],[450,3133,3135,3138,3141,3144,3147,3150,3153],{"className":3134},[462],[450,3136],{"className":3137,"style":1395},[466],[450,3139,2440],{"className":3140},[471],[450,3142,2302],{"className":3143,"style":2301},[471,1114],[450,3145,2415],{"className":3146},[471],[450,3148],{"className":3149,"style":1154},[1153],[450,3151,1216],{"className":3152},[1158],[450,3154],{"className":3155,"style":1154},[1153],[450,3157,3159,3162,3165],{"className":3158},[462],[450,3160],{"className":3161,"style":1271},[466],[450,3163,1473],{"className":3164},[471],[450,3166,3168,3171],{"className":3167},[471],[450,3169,1192],{"className":3170,"style":1191},[471,1114],[450,3172,3174],{"className":3173},[1258],[450,3175,3177],{"className":3176},[1262],[450,3178,3180],{"className":3179},[1266],[450,3181,3183],{"className":3182,"style":1271},[1270],[450,3184,3185,3188],{"style":1274},[450,3186],{"className":3187,"style":1279},[1278],[450,3189,3191],{"className":3190},[1283,1284,1285,1286],[450,3192,3194],{"className":3193},[471,1286],[450,3195,1293],{"className":3196},[471,1286],[381,3198,3199],{},"Not every problem is so cooperative. Approximability varies wildly:",[390,3201,3202,3302,3320],{},[393,3203,3204,3205,3208,3209,3246,3247,3287,3288,446,3291,3294,3295],{},"Some problems admit a ",[385,3206,3207],{},"polynomial-time approximation scheme"," (PTAS): for any\n",[450,3210,3212],{"className":3211},[453],[450,3213,3215,3236],{"className":3214,"ariaHidden":458},[457],[450,3216,3218,3222,3226,3229,3233],{"className":3217},[462],[450,3219],{"className":3220,"style":3221},[466],"height:0.5782em;vertical-align:-0.0391em;",[450,3223,3225],{"className":3224},[471,1114],"ε",[450,3227],{"className":3228,"style":1154},[1153],[450,3230,3232],{"className":3231},[1158],">",[450,3234],{"className":3235,"style":1154},[1153],[450,3237,3239,3242],{"className":3238},[462],[450,3240],{"className":3241,"style":1169},[466],[450,3243,3245],{"className":3244},[471],"0"," a ",[450,3248,3250],{"className":3249},[453],[450,3251,3253,3275],{"className":3252,"ariaHidden":458},[457],[450,3254,3256,3259,3262,3265,3268,3272],{"className":3255},[462],[450,3257],{"className":3258,"style":1395},[466],[450,3260,2068],{"className":3261},[1875],[450,3263,1173],{"className":3264},[471],[450,3266],{"className":3267,"style":1233},[1153],[450,3269,3271],{"className":3270},[1237],"+",[450,3273],{"className":3274,"style":1233},[1153],[450,3276,3278,3281,3284],{"className":3277},[462],[450,3279],{"className":3280,"style":1395},[466],[450,3282,3225],{"className":3283},[471,1114],[450,3285,2087],{"className":3286},[1957],"-approximation in polynomial time: you\ncan get as close to optimal as you like, paying in running time.\n",[385,3289,3290],{},"Euclidean TSP",[385,3292,3293],{},"Knapsack"," are of this kind.",[2685,3296,3297],{},[406,3298,1473],{"href":3299,"ariaDescribedBy":3300,"dataFootnoteRef":376,"id":3301},"#user-content-fn-clrs-ptas",[2691],"user-content-fnref-clrs-ptas",[393,3303,3304,3305,416],{},"Some have a fixed best-possible constant ratio, like vertex cover's ",[450,3306,3308],{"className":3307},[453],[450,3309,3311],{"className":3310,"ariaHidden":458},[457],[450,3312,3314,3317],{"className":3313},[462],[450,3315],{"className":3316,"style":1169},[466],[450,3318,1473],{"className":3319},[471],[393,3321,3322,3323,3326,3327,3364,3365,3368,3369,3372,3373,3391],{},"Some are ",[385,3324,3325],{},"inapproximable",": unless ",[450,3328,3330],{"className":3329},[453],[450,3331,3333,3352],{"className":3332,"ariaHidden":458},[457],[450,3334,3336,3339,3343,3346,3349],{"className":3335},[462],[450,3337],{"className":3338,"style":467},[466],[450,3340,3342],{"className":3341},[471,475],"P",[450,3344],{"className":3345,"style":1154},[1153],[450,3347,1460],{"className":3348},[1158],[450,3350],{"className":3351,"style":1154},[1153],[450,3353,3355,3358],{"className":3354},[462],[450,3356],{"className":3357,"style":467},[466],[450,3359,3361],{"className":3360},[471],[450,3362,476],{"className":3363},[471,475],", no\npolynomial-time algorithm achieves ",[419,3366,3367],{},"any"," constant ratio. ",[385,3370,3371],{},"General TSP","\n(without triangle inequality) is the classic example: even approximating it\nwithin any factor is ",[450,3374,3376],{"className":3375},[453],[450,3377,3379],{"className":3378,"ariaHidden":458},[457],[450,3380,3382,3385],{"className":3381},[462],[450,3383],{"className":3384,"style":467},[466],[450,3386,3388],{"className":3387},[471],[450,3389,476],{"className":3390},[471,475],"-hard.",[381,3393,3394,3395,3398,3399,3402,3403,3418,3419,3422,3423,3426,3427,3480],{},"When the triangle inequality ",[419,3396,3397],{},"does"," hold (the ",[385,3400,3401],{},"metric"," case), a clean\n",[450,3404,3406],{"className":3405},[453],[450,3407,3409],{"className":3408,"ariaHidden":458},[457],[450,3410,3412,3415],{"className":3411},[462],[450,3413],{"className":3414,"style":1169},[466],[450,3416,1473],{"className":3417},[471],"-approximation falls out of the ",[406,3420,3421],{"href":170},"minimum spanning tree",": double every MST\nedge to get an Eulerian multigraph, walk an Euler tour, and ",[385,3424,3425],{},"shortcut"," past\nalready-visited vertices. The shortcuts only shorten the walk (triangle\ninequality), so the tour costs at most ",[450,3428,3430],{"className":3429},[453],[450,3431,3433,3461],{"className":3432,"ariaHidden":458},[457],[450,3434,3436,3439,3442,3445,3452,3455,3458],{"className":3435},[462],[450,3437],{"className":3438,"style":1206},[466],[450,3440,1473],{"className":3441},[471],[450,3443],{"className":3444,"style":1915},[1153],[450,3446,3448],{"className":3447},[471,2109],[450,3449,3451],{"className":3450},[471],"MST",[450,3453],{"className":3454,"style":1154},[1153],[450,3456,1216],{"className":3457},[1158],[450,3459],{"className":3460,"style":1154},[1153],[450,3462,3464,3467,3470,3473],{"className":3463},[462],[450,3465],{"className":3466,"style":1187},[466],[450,3468,1473],{"className":3469},[471],[450,3471],{"className":3472,"style":1915},[1153],[450,3474,3476],{"className":3475},[471,2109],[450,3477,3479],{"className":3478},[471],"OPT",", since\nthe MST is no heavier than the optimal tour with one edge removed.",[559,3482,3484,3708],{"className":3483},[562,563],[565,3485,3489],{"xmlns":567,"width":3486,"height":3487,"viewBox":3488},"355.062","131.081","-75 -75 266.296 98.311",[572,3490,3491,3503,3515,3527,3539,3551,3554,3587,3617,3628,3639,3650,3661,3672,3675],{"stroke":574,"style":575},[572,3492,3493,3496],{"fill":1608},[580,3494],{"d":3495},"M-51.177-53.576a7.113 7.113 0 1 0-14.226 0 7.113 7.113 0 0 0 14.226 0Zm-7.113 0",[572,3497,3499],{"transform":3498},"translate(-2.45 -54.968)",[580,3500],{"d":3501,"fill":574,"stroke":574,"className":3502,"style":595},"M-56.668 3.431Q-57.064 3.431-57.350 3.227Q-57.635 3.022-57.782 2.688Q-57.930 2.354-57.930 1.963Q-57.930 1.528-57.756 1.067Q-57.582 0.605-57.270 0.214Q-56.958-0.177-56.548-0.412Q-56.137-0.647-55.697-0.647Q-55.429-0.647-55.212-0.509Q-54.994-0.370-54.862-0.124Q-54.823-0.274-54.715-0.370Q-54.607-0.467-54.467-0.467Q-54.344-0.467-54.260-0.394Q-54.177-0.322-54.177-0.199Q-54.177-0.146-54.186-0.115L-54.805 2.376Q-54.862 2.574-54.862 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",[450,3729,3731],{"className":3730},[453],[450,3732,3734,3747,3774],{"className":3733,"ariaHidden":458},[457],[450,3735,3737,3741,3744],{"className":3736},[462],[450,3738],{"className":3739,"style":3740},[466],"height:0.7719em;vertical-align:-0.136em;",[450,3742,1216],{"className":3743},[1158],[450,3745],{"className":3746,"style":1154},[1153],[450,3748,3750,3753,3756,3759,3765,3768,3771],{"className":3749},[462],[450,3751],{"className":3752,"style":1206},[466],[450,3754,1473],{"className":3755},[471],[450,3757],{"className":3758,"style":1915},[1153],[450,3760,3762],{"className":3761},[471,2109],[450,3763,3451],{"className":3764},[471],[450,3766],{"className":3767,"style":1154},[1153],[450,3769,1216],{"className":3770},[1158],[450,3772],{"className":3773,"style":1154},[1153],[450,3775,3777,3780,3783,3786],{"className":3776},[462],[450,3778],{"className":3779,"style":1187},[466],[450,3781,1473],{"className":3782},[471],[450,3784],{"className":3785,"style":1915},[1153],[450,3787,3789],{"className":3788},[471,2109],[450,3790,3479],{"className":3791},[471],[1070,3793,3795],{"id":3794},"heuristics-and-local-search","Heuristics and local search",[381,3797,3798,3799,3801,3802,3805,3806,3809,3810,3813],{},"When a provable ratio is out of reach or simply unnecessary, we turn to\n",[385,3800,537],{},", strategies that are usually good but carry no worst-case\npromise. The most general is ",[385,3803,3804],{},"local search",": start from some feasible\nsolution and repeatedly apply a small modification, a ",[419,3807,3808],{},"move",", that improves\nthe objective, stopping at a ",[385,3811,3812],{},"local optimum"," where no single move helps.",[381,3815,3816,3817,3820],{},"For the traveling salesman, the famous ",[385,3818,3819],{},"2-opt"," move deletes two edges of the\ncurrent tour and reconnects the pieces the other way; iterating it untangles\ncrossings and converges to short, though not always optimal, tours.",[559,3822,3824,4037],{"className":3823},[562,563],[565,3825,3829],{"xmlns":567,"width":3826,"height":3827,"viewBox":3828},"374.340","135.895","-75 -75 280.755 101.921",[572,3830,3831,3842,3854,3866,3878,3881,3885,3930,3945,3956,3967,3978,3989,3992],{"stroke":574,"style":575},[572,3832,3833,3836],{"fill":1608},[580,3834],{"d":3835},"M-36.623-56.301a6.402 6.402 0 1 0-12.803 0 6.402 6.402 0 0 0 12.803 0Zm-6.402 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characteristic failure mode: it can stall in a local optimum far\nfrom the global one. The standard escapes are ",[419,4045,4046],{},"metaheuristics"," that\noccasionally accept worsening moves to climb out of bad valleys:",[390,4049,4050,4056,4062],{},[393,4051,4052,4055],{},[385,4053,4054],{},"Simulated annealing"," accepts uphill moves with a probability that cools\nover time, mimicking the physics of slowly freezing metal.",[393,4057,4058,4061],{},[385,4059,4060],{},"Tabu search"," forbids recently-visited solutions to avoid cycling back.",[393,4063,4064,4067],{},[385,4065,4066],{},"Genetic algorithms"," evolve a population of solutions by recombination and\nmutation.",[559,4069,4071,4263],{"className":4070},[562,563],[565,4072,4076],{"xmlns":567,"width":4073,"height":4074,"viewBox":4075},"397.571","150.363","-75 -75 298.178 112.772",[572,4077,4078,4081,4084,4091,4094,4097,4112,4115,4120,4171,4176,4197,4206],{"stroke":574,"style":575},[580,4079],{"fill":582,"d":4080},"M-44.903 21.414v-79.437",[580,4082],{"d":4083},"m-44.903-60.529-1.35 3.585 1.35-1.18 1.351 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3.130 21.103Q3.373 21.260 3.650 21.260Q3.978 21.260 4.231 21.059Q4.484 20.857 4.617 20.539Q4.750 20.221 4.750 19.903Q4.750 19.674 4.685 19.445Q4.620 19.216 4.492 19.018Q4.364 18.820 4.169 18.700Q3.974 18.581 3.742 18.581Q3.448 18.581 3.180 18.710Q2.911 18.840 2.754 19.083",[594],[572,4258,4259],{"transform":4210},[580,4260],{"d":4261,"fill":3883,"stroke":3883,"className":4262,"style":946},"M6.246 19.879Q6.246 19.558 6.371 19.269Q6.496 18.980 6.722 18.757Q6.947 18.533 7.243 18.413Q7.538 18.293 7.856 18.293Q8.184 18.293 8.446 18.393Q8.707 18.492 8.883 18.674Q9.059 18.857 9.153 19.115Q9.247 19.373 9.247 19.705Q9.247 19.797 9.165 19.818L6.910 19.818L6.910 19.879Q6.910 20.467 7.193 20.850Q7.477 21.233 8.044 21.233Q8.366 21.233 8.634 21.040Q8.902 20.847 8.991 20.532Q8.998 20.491 9.073 20.477L9.165 20.477Q9.247 20.501 9.247 20.573Q9.247 20.580 9.241 20.607Q9.128 21.004 8.757 21.243Q8.386 21.482 7.962 21.482Q7.525 21.482 7.125 21.274Q6.725 21.065 6.486 20.698Q6.246 20.331 6.246 19.879M6.916 19.609L8.731 19.609Q8.731 19.332 8.634 19.080Q8.536 18.827 8.338 18.671Q8.140 18.516 7.856 18.516Q7.579 18.516 7.366 18.674Q7.152 18.833 7.034 19.088Q6.916 19.343 6.916 19.609",[594],[1064,4264,4266],{"className":4265},[1067],"Local search descends to the nearest valley and stalls at a local optimum; a metaheuristic must accept an uphill move to escape and reach the global optimum.",[381,4268,4269,4270],{},"Heuristics are the workhorses of industry precisely because they are flexible\nand fast. Their cost is the loss of guarantees: you rarely know how far from\noptimal you landed. The honest practice, Skiena stresses, is to test against\nknown optima on small instances and against lower bounds on large ones.",[2685,4271,4272],{},[406,4273,4277],{"href":4274,"ariaDescribedBy":4275,"dataFootnoteRef":376,"id":4276},"#user-content-fn-skiena-heur",[2691],"user-content-fnref-skiena-heur","3",[1070,4279,4281],{"id":4280},"exact-exponential-methods-branch-and-bound","Exact exponential methods: branch and bound",[381,4283,4284,4285,4288,4289,4292,4293,416],{},"Sometimes you genuinely need the ",[419,4286,4287],{},"optimal"," answer and the instances are small\nenough to afford exponential time, provided it is spent wisely. The key idea is to\nsearch the space of solutions as a tree while ",[385,4290,4291],{},"pruning"," subtrees that\nprovably cannot beat the best solution found so far. This is ",[385,4294,4295],{},[406,4296,4297],{"href":303},"branch and\nbound",[381,4299,4300,4301,4304,4305,4324,4325,4343,4344,4347,4348,4351,4352,4355],{},"The method interleaves two operations. ",[419,4302,4303],{},"Branching"," splits the problem into\nsubproblems (e.g. ",[1476,4306,4307,4308,4323],{},"vertex ",[450,4309,4311],{"className":4310},[453],[450,4312,4314],{"className":4313,"ariaHidden":458},[457],[450,4315,4317,4320],{"className":4316},[462],[450,4318],{"className":4319,"style":2370},[466],[450,4321,2353],{"className":4322,"style":2352},[471,1114]," is in the cover"," vs. ",[1476,4326,4327,4342],{},[450,4328,4330],{"className":4329},[453],[450,4331,4333],{"className":4332,"ariaHidden":458},[457],[450,4334,4336,4339],{"className":4335},[462],[450,4337],{"className":4338,"style":2370},[466],[450,4340,2353],{"className":4341,"style":2352},[471,1114]," is out","), forming a\nsearch tree. ",[419,4345,4346],{},"Bounding"," computes, for each subproblem, an optimistic estimate,\na ",[385,4349,4350],{},"bound",", on the best solution reachable within it. If that optimistic\nestimate is already no better than the best complete solution we have already\nfound (the ",[385,4353,4354],{},"incumbent","), the entire subtree is discarded unexplored.",[1090,4357,4359],{"type":4358},"remark",[381,4360,4361,4364,4365,4381,4382,4397,4398,4431,4432,4435],{},[385,4362,4363],{},"Remark (The pruning rule)."," For a minimization problem, maintain the incumbent cost\n",[450,4366,4368],{"className":4367},[453],[450,4369,4371],{"className":4370,"ariaHidden":458},[457],[450,4372,4374,4377],{"className":4373},[462],[450,4375],{"className":4376,"style":1187},[466],[450,4378,4380],{"className":4379,"style":2301},[471,1114],"U"," (best solution so far). At a subproblem with optimistic lower bound ",[450,4383,4385],{"className":4384},[453],[450,4386,4388],{"className":4387,"ariaHidden":458},[457],[450,4389,4391,4394],{"className":4390},[462],[450,4392],{"className":4393,"style":1187},[466],[450,4395,1975],{"className":4396},[471,1114],",\nif ",[450,4399,4401],{"className":4400},[453],[450,4402,4404,4422],{"className":4403,"ariaHidden":458},[457],[450,4405,4407,4410,4413,4416,4419],{"className":4406},[462],[450,4408],{"className":4409,"style":1206},[466],[450,4411,1975],{"className":4412},[471,1114],[450,4414],{"className":4415,"style":1154},[1153],[450,4417,1159],{"className":4418},[1158],[450,4420],{"className":4421,"style":1154},[1153],[450,4423,4425,4428],{"className":4424},[462],[450,4426],{"className":4427,"style":1187},[466],[450,4429,4380],{"className":4430,"style":2301},[471,1114],", ",[419,4433,4434],{},"prune"," — nothing in this subtree can improve on the incumbent.",[381,4437,4438,4439,4457],{},"The worst case is still exponential; branch and bound does not repeal\n",[450,4440,4442],{"className":4441},[453],[450,4443,4445],{"className":4444,"ariaHidden":458},[457],[450,4446,4448,4451],{"className":4447},[462],[450,4449],{"className":4450,"style":467},[466],[450,4452,4454],{"className":4453},[471],[450,4455,476],{"className":4456},[471,475],"-hardness. But on real instances a good bound can prune away the\noverwhelming majority of the tree, making problems with thousands of variables\nroutinely solvable. The quality of the bound is everything: a tight bound (often\nfrom a relaxation such as linear programming) prunes aggressively; a loose one\nleaves a near-complete exponential search. Modern integer-programming solvers\nare, at heart, exquisitely engineered branch-and-bound engines.",[559,4459,4461,4679],{"className":4460},[562,563],[565,4462,4466],{"xmlns":567,"width":4463,"height":4464,"viewBox":4465},"344.191","170.977","-75 -75 258.143 128.233",[572,4467,4468,4471,4476,4483,4486,4491,4494,4498,4519,4522,4525,4544,4547,4551,4554,4557,4560,4563,4566,4569,4602,4629,4652],{"stroke":574,"style":575},[580,4469],{"fill":582,"d":4470},"M74.248-60.689c0-6.286-5.096-11.381-11.382-11.381s-11.38 5.095-11.38 11.381 5.095 11.381 11.38 11.381 11.382-5.095 11.382-11.38ZM5.961-20.855c0-6.286-5.095-11.38-11.38-11.38-6.287 0-11.382 5.094-11.382 11.38S-11.706-9.474-5.42-9.474s11.381-5.095 11.381-11.38ZM142.534-20.855c0-6.286-5.095-11.38-11.381-11.38s-11.381 5.094-11.381 11.38 5.095 11.381 11.38 11.381c6.287 0 11.382-5.095 11.382-11.38Zm-11.381 0",[572,4472,4473],{"fill":1686,"stroke":1687,"style":578},[580,4474],{"d":4475},"M-28.182 18.979c0-6.286-5.096-11.381-11.382-11.381s-11.38 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95.087 13.062L94.821 13.336Q94.800 13.356 94.773 13.356L94.725 13.356Q94.701 13.356 94.674 13.329Q94.646 13.302 94.646 13.281M97.770 11.805Q97.770 11.463 97.905 11.164Q98.040 10.865 98.280 10.641Q98.519 10.417 98.837 10.292Q99.155 10.167 99.486 10.167Q99.931 10.167 100.331 10.383Q100.730 10.598 100.965 10.976Q101.199 11.353 101.199 11.805Q101.199 12.146 101.057 12.430Q100.915 12.714 100.671 12.921Q100.426 13.127 100.117 13.242Q99.808 13.356 99.486 13.356Q99.056 13.356 98.654 13.155Q98.252 12.953 98.011 12.601Q97.770 12.249 97.770 11.805M99.486 13.107Q100.088 13.107 100.312 12.729Q100.536 12.351 100.536 11.719Q100.536 11.107 100.301 10.748Q100.067 10.390 99.486 10.390Q98.434 10.390 98.434 11.719Q98.434 12.351 98.659 12.729Q98.885 13.107 99.486 13.107M103.461 13.288L101.858 13.288L101.858 13.008Q102.084 13.008 102.233 12.974Q102.381 12.939 102.381 12.799L102.381 9.180Q102.381 8.910 102.274 8.848Q102.166 8.787 101.858 8.787L101.858 8.506L102.935 8.431L102.935 12.799Q102.935 12.936 103.085 12.972Q103.236 13.008 103.461 13.008L103.461 13.288M104.630 12.454L104.630 10.950Q104.630 10.680 104.523 10.619Q104.415 10.557 104.104 10.557L104.104 10.277L105.211 10.202L105.211 12.434L105.211 12.454Q105.211 12.734 105.263 12.878Q105.314 13.021 105.456 13.078Q105.598 13.134 105.885 13.134Q106.138 13.134 106.343 12.994Q106.548 12.854 106.664 12.628Q106.780 12.403 106.780 12.153L106.780 10.950Q106.780 10.680 106.673 10.619Q106.565 10.557 106.254 10.557L106.254 10.277L107.361 10.202L107.361 12.615Q107.361 12.806 107.414 12.888Q107.467 12.970 107.568 12.989Q107.669 13.008 107.884 13.008L107.884 13.288L106.808 13.356L106.808 12.792Q106.698 12.974 106.553 13.097Q106.408 13.220 106.221 13.288Q106.035 13.356 105.833 13.356Q104.630 13.356 104.630 12.454M108.999 12.447L108.999 10.550L108.359 10.550L108.359 10.328Q108.677 10.328 108.894 10.118Q109.111 9.908 109.212 9.598Q109.313 9.289 109.313 8.981L109.580 8.981L109.580 10.270L110.656 10.270L110.656 10.550L109.580 10.550L109.580 12.434Q109.580 12.710 109.684 12.909Q109.788 13.107 110.048 13.107Q110.205 13.107 110.311 13.003Q110.417 12.898 110.467 12.745Q110.516 12.591 110.516 12.434L110.516 12.020L110.783 12.020L110.783 12.447Q110.783 12.673 110.684 12.883Q110.584 13.093 110.400 13.225Q110.215 13.356 109.986 13.356Q109.549 13.356 109.274 13.119Q108.999 12.881 108.999 12.447M113.209 13.288L111.658 13.288L111.658 13.008Q111.883 13.008 112.032 12.974Q112.181 12.939 112.181 12.799L112.181 10.950Q112.181 10.762 112.133 10.678Q112.085 10.595 111.988 10.576Q111.890 10.557 111.678 10.557L111.678 10.277L112.734 10.202L112.734 12.799Q112.734 12.939 112.866 12.974Q112.998 13.008 113.209 13.008L113.209 13.288M111.938 8.981Q111.938 8.810 112.061 8.691Q112.184 8.571 112.355 8.571Q112.522 8.571 112.645 8.691Q112.769 8.810 112.769 8.981Q112.769 9.156 112.645 9.279Q112.522 9.402 112.355 9.402Q112.184 9.402 112.061 9.279Q111.938 9.156 111.938 8.981M113.814 11.805Q113.814 11.463 113.949 11.164Q114.084 10.865 114.324 10.641Q114.563 10.417 114.881 10.292Q115.199 10.167 115.530 10.167Q115.975 10.167 116.374 10.383Q116.774 10.598 117.009 10.976Q117.243 11.353 117.243 11.805Q117.243 12.146 117.101 12.430Q116.959 12.714 116.715 12.921Q116.470 13.127 116.161 13.242Q115.852 13.356 115.530 13.356Q115.100 13.356 114.698 13.155Q114.296 12.953 114.055 12.601Q113.814 12.249 113.814 11.805M115.530 13.107Q116.132 13.107 116.356 12.729Q116.580 12.351 116.580 11.719Q116.580 11.107 116.345 10.748Q116.111 10.390 115.530 10.390Q114.478 10.390 114.478 11.719Q114.478 12.351 114.703 12.729Q114.929 13.107 115.530 13.107M119.519 13.288L117.885 13.288L117.885 13.008Q118.114 13.008 118.263 12.974Q118.412 12.939 118.412 12.799L118.412 10.950Q118.412 10.680 118.304 10.619Q118.196 10.557 117.885 10.557L117.885 10.277L118.945 10.202L118.945 10.851Q119.116 10.543 119.420 10.372Q119.724 10.202 120.069 10.202Q120.575 10.202 120.859 10.425Q121.143 10.649 121.143 11.145L121.143 12.799Q121.143 12.936 121.291 12.972Q121.440 13.008 121.665 13.008L121.665 13.288L120.035 13.288L120.035 13.008Q120.264 13.008 120.413 12.974Q120.561 12.939 120.561 12.799L120.561 11.159Q120.561 10.824 120.442 10.624Q120.322 10.424 120.008 10.424Q119.738 10.424 119.504 10.560Q119.269 10.697 119.131 10.931Q118.993 11.165 118.993 11.439L118.993 12.799Q118.993 12.936 119.143 12.972Q119.293 13.008 119.519 13.008L119.519 13.288M122.253 13.281L122.253 12.218Q122.253 12.194 122.281 12.167Q122.308 12.140 122.332 12.140L122.441 12.140Q122.506 12.140 122.520 12.198Q122.616 12.632 122.862 12.883Q123.108 13.134 123.521 13.134Q123.863 13.134 124.116 13.001Q124.369 12.868 124.369 12.560Q124.369 12.403 124.275 12.288Q124.181 12.174 124.043 12.105Q123.904 12.037 123.737 11.999L123.156 11.900Q122.800 11.832 122.527 11.611Q122.253 11.391 122.253 11.049Q122.253 10.800 122.364 10.625Q122.476 10.451 122.662 10.352Q122.848 10.253 123.063 10.210Q123.279 10.167 123.521 10.167Q123.935 10.167 124.215 10.349L124.431 10.174Q124.441 10.171 124.448 10.169Q124.455 10.167 124.465 10.167L124.516 10.167Q124.543 10.167 124.567 10.191Q124.591 10.215 124.591 10.243L124.591 11.090Q124.591 11.111 124.567 11.138Q124.543 11.165 124.516 11.165L124.403 11.165Q124.376 11.165 124.350 11.140Q124.325 11.114 124.325 11.090Q124.325 10.854 124.219 10.690Q124.113 10.526 123.930 10.444Q123.747 10.362 123.515 10.362Q123.186 10.362 122.930 10.465Q122.674 10.567 122.674 10.844Q122.674 11.039 122.857 11.148Q123.040 11.258 123.269 11.299L123.843 11.405Q124.089 11.453 124.302 11.581Q124.516 11.709 124.653 11.912Q124.790 12.116 124.790 12.365Q124.790 12.878 124.424 13.117Q124.058 13.356 123.521 13.356Q123.026 13.356 122.694 13.062L122.428 13.336Q122.407 13.356 122.380 13.356L122.332 13.356Q122.308 13.356 122.281 13.329Q122.253 13.302 122.253 13.281",[594],[1064,4680,4682,4683,4698,4699,416],{"className":4681},[1067],"Branch and bound: split on a variable, bound each subtree, and prune any whose optimistic bound ",[450,4684,4686],{"className":4685},[453],[450,4687,4689],{"className":4688,"ariaHidden":458},[457],[450,4690,4692,4695],{"className":4691},[462],[450,4693],{"className":4694,"style":1187},[466],[450,4696,1975],{"className":4697},[471,1114]," is no better than the incumbent ",[450,4700,4702],{"className":4701},[453],[450,4703,4705],{"className":4704,"ariaHidden":458},[457],[450,4706,4708,4711],{"className":4707},[462],[450,4709],{"className":4710,"style":1187},[466],[450,4712,4380],{"className":4713,"style":2301},[471,1114],[1070,4715,4717],{"id":4716},"exploiting-special-structure","Exploiting special structure",[381,4719,4720,4721,4739,4740,4743,4744,4747],{},"The final and most underrated strategy is to remember that\n",[450,4722,4724],{"className":4723},[453],[450,4725,4727],{"className":4726,"ariaHidden":458},[457],[450,4728,4730,4733],{"className":4729},[462],[450,4731],{"className":4732,"style":467},[466],[450,4734,4736],{"className":4735},[471],[450,4737,476],{"className":4738},[471,475],"-hardness is a worst case over ",[419,4741,4742],{},"all"," inputs, and your inputs may\nnot be the worst. Many hard problems become ",[419,4745,4746],{},"easy"," when restricted to the\nstructured instances that arise in practice.",[390,4749,4750,4787,5032],{},[393,4751,4752,4755,4756,4774,4775,4778,4779,4782,4783,4786],{},[385,4753,4754],{},"Restricted graph classes."," Problems that are ",[450,4757,4759],{"className":4758},[453],[450,4760,4762],{"className":4761,"ariaHidden":458},[457],[450,4763,4765,4768],{"className":4764},[462],[450,4766],{"className":4767,"style":467},[466],[450,4769,4771],{"className":4770},[471],[450,4772,476],{"className":4773},[471,475],"-hard on general\ngraphs frequently admit polynomial (even linear) algorithms on ",[385,4776,4777],{},"trees",",\non ",[385,4780,4781],{},"bipartite"," graphs, or on ",[385,4784,4785],{},"planar"," graphs. Independent Set, hard in\ngeneral, falls to a simple greedy\u002Fdynamic program on trees.",[393,4788,4789,4792,4793,4881,4882,4897,4898,4913,4914,4929,4930,4933,4934,5007,5008,5023,5024],{},[385,4790,4791],{},"Bounded parameters."," A problem may be solvable in time\n",[450,4794,4796],{"className":4795},[453],[450,4797,4799,4830],{"className":4798,"ariaHidden":458},[457],[450,4800,4802,4805,4810,4813,4818,4821,4824,4827],{"className":4801},[462],[450,4803],{"className":4804,"style":1395},[466],[450,4806,4809],{"className":4807,"style":4808},[471,1114],"margin-right:0.1076em;","f",[450,4811,2068],{"className":4812},[1875],[450,4814,4817],{"className":4815,"style":4816},[471,1114],"margin-right:0.0315em;","k",[450,4819,2087],{"className":4820},[1957],[450,4822],{"className":4823,"style":1233},[1153],[450,4825,1238],{"className":4826},[1237],[450,4828],{"className":4829,"style":1233},[1153],[450,4831,4833,4837],{"className":4832},[462],[450,4834],{"className":4835,"style":4836},[466],"height:0.888em;",[450,4838,4840,4844],{"className":4839},[471],[450,4841,4843],{"className":4842},[471,1114],"n",[450,4845,4847],{"className":4846},[1258],[450,4848,4850],{"className":4849},[1262],[450,4851,4853],{"className":4852},[1266],[450,4854,4856],{"className":4855,"style":4836},[1270],[450,4857,4858,4861],{"style":1274},[450,4859],{"className":4860,"style":1279},[1278],[450,4862,4864],{"className":4863},[1283,1284,1285,1286],[450,4865,4867,4872,4875,4878],{"className":4866},[471,1286],[450,4868,4871],{"className":4869,"style":4870},[471,1114,1286],"margin-right:0.0278em;","O",[450,4873,2068],{"className":4874},[1875,1286],[450,4876,1173],{"className":4877},[471,1286],[450,4879,2087],{"className":4880},[1957,1286],", where ",[450,4883,4885],{"className":4884},[453],[450,4886,4888],{"className":4887,"ariaHidden":458},[457],[450,4889,4891,4894],{"className":4890},[462],[450,4892],{"className":4893,"style":467},[466],[450,4895,4817],{"className":4896,"style":4816},[471,1114]," is some small parameter of the instance (the\nsolution size, the treewidth, the number of constraints). The exponential\nblow-up is confined to ",[450,4899,4901],{"className":4900},[453],[450,4902,4904],{"className":4903,"ariaHidden":458},[457],[450,4905,4907,4910],{"className":4906},[462],[450,4908],{"className":4909,"style":467},[466],[450,4911,4817],{"className":4912,"style":4816},[471,1114],", so if ",[450,4915,4917],{"className":4916},[453],[450,4918,4920],{"className":4919,"ariaHidden":458},[457],[450,4921,4923,4926],{"className":4922},[462],[450,4924],{"className":4925,"style":467},[466],[450,4927,4817],{"className":4928,"style":4816},[471,1114]," is small the algorithm is fast. This is\nthe domain of ",[385,4931,4932],{},"fixed-parameter tractability",": Vertex Cover, for instance, is\nsolvable in ",[450,4935,4937],{"className":4936},[453],[450,4938,4940,4995],{"className":4939,"ariaHidden":458},[457],[450,4941,4943,4947,4950,4953,4986,4989,4992],{"className":4942},[462],[450,4944],{"className":4945,"style":4946},[466],"height:1.0991em;vertical-align:-0.25em;",[450,4948,4871],{"className":4949,"style":4870},[471,1114],[450,4951,2068],{"className":4952},[1875],[450,4954,4956,4959],{"className":4955},[471],[450,4957,1473],{"className":4958},[471],[450,4960,4962],{"className":4961},[1258],[450,4963,4965],{"className":4964},[1262],[450,4966,4968],{"className":4967},[1266],[450,4969,4972],{"className":4970,"style":4971},[1270],"height:0.8491em;",[450,4973,4974,4977],{"style":1274},[450,4975],{"className":4976,"style":1279},[1278],[450,4978,4980],{"className":4979},[1283,1284,1285,1286],[450,4981,4983],{"className":4982},[471,1286],[450,4984,4817],{"className":4985,"style":4816},[471,1114,1286],[450,4987],{"className":4988,"style":1233},[1153],[450,4990,1238],{"className":4991},[1237],[450,4993],{"className":4994,"style":1233},[1153],[450,4996,4998,5001,5004],{"className":4997},[462],[450,4999],{"className":5000,"style":1395},[466],[450,5002,4843],{"className":5003},[471,1114],[450,5005,2087],{"className":5006},[1957]," for covers of size ",[450,5009,5011],{"className":5010},[453],[450,5012,5014],{"className":5013,"ariaHidden":458},[457],[450,5015,5017,5020],{"className":5016},[462],[450,5018],{"className":5019,"style":467},[466],[450,5021,4817],{"className":5022,"style":4816},[471,1114],", practical whenever the\ncover is small even if the graph is huge.",[2685,5025,5026],{},[406,5027,5031],{"href":5028,"ariaDescribedBy":5029,"dataFootnoteRef":376,"id":5030},"#user-content-fn-erickson-struct",[2691],"user-content-fnref-erickson-struct","4",[393,5033,5034,5037,5038,5060,5061,5063,5064,5067],{},[385,5035,5036],{},"Pseudo-polynomial algorithms."," Numeric problems like ",[450,5039,5041],{"className":5040},[453],[450,5042,5044],{"className":5043,"ariaHidden":458},[457],[450,5045,5047,5050],{"className":5046},[462],[450,5048],{"className":5049,"style":467},[466],[450,5051,5053],{"className":5052},[2104,2105],[450,5054,5056],{"className":5055},[471,2109],[450,5057,5059],{"className":5058},[471],"Subset-Sum"," and\n",[385,5062,3293],{}," have dynamic programs running in time polynomial in the\n",[419,5065,5066],{},"numeric values",", fast when the numbers are modest, exponential only because\nvalues can be exponentially large in their bit-length.",[381,5069,5070],{},"The lesson is to look hard at the instances you must actually solve before\ndeclaring defeat. Hardness in the worst case is fully compatible with easiness\nin your case.",[1070,5072,5074],{"id":5073},"choosing-a-strategy","Choosing a strategy",[381,5076,5077],{},"These responses are not rivals so much as a toolkit; the right choice depends on\nwhat you can tolerate.",[390,5079,5080,5092,5108,5118],{},[393,5081,5082,5083,5085,5086,5089,5090,416],{},"Need a ",[419,5084,1083],{}," and can accept ",[1476,5087,5088],{},"near-optimal","? Reach for an\n",[385,5091,1087],{},[393,5093,5094,5095,446,5098,5101,5102,2374,5105,5107],{},"Need ",[419,5096,5097],{},"speed",[419,5099,5100],{},"flexibility"," and can live without guarantees? Use a\n",[385,5103,5104],{},"heuristic",[385,5106,3804],{},", validated empirically.",[393,5109,5110,5111,5114,5115,5117],{},"Need the ",[419,5112,5113],{},"exact"," optimum on instances of modest size? Invest in ",[385,5116,4297],{}," with the tightest bound you can compute.",[393,5119,5120,5121,5124,5125,5128],{},"Do your real instances have ",[419,5122,5123],{},"structure",", such as small parameters, special graph\nshape, or modest numbers? ",[385,5126,5127],{},"Exploit it",", possibly turning the problem\npolynomial outright.",[1070,5130,5132],{"id":5131},"final-thoughts","Final thoughts",[381,5134,5135,5136,5139,5140,5143,5144,5147,5148,5166,5167,5170],{},"Step back and the whole course resolves into a single arc. We learned to ",[419,5137,5138],{},"analyze","\n(asymptotics, recurrences, invariants), to ",[419,5141,5142],{},"design"," across a small repertoire of\nparadigms (divide and conquer, dynamic programming, greedy, graph search, network\nflow), and finally to ",[419,5145,5146],{},"recognize the limits of design"," through reductions and\n",[450,5149,5151],{"className":5150},[453],[450,5152,5154],{"className":5153,"ariaHidden":458},[457],[450,5155,5157,5160],{"className":5156},[462],[450,5158],{"className":5159,"style":467},[466],[450,5161,5163],{"className":5162},[471],[450,5164,476],{"className":5165},[471,475],"-completeness. This last lesson closes the loop: when the limit is\nreal, you do not give up — you change the question, trading exactness, optimality,\nor generality for tractability, and doing so ",[419,5168,5169],{},"honestly",", with a clear statement of\nwhat you gave up.",[381,5172,5173,5174,5177],{},"The paradigms compose. A branch-and-bound solver leans on a relaxation (often\nlinear programming) for its bound; an approximation proof leans on a combinatorial\nlower bound like a matching; a special-case algorithm is frequently just dynamic\nprogramming rediscovered on a tree. The reduction habit — ",[419,5175,5176],{},"map my problem onto one\nI already understand"," — is the same instinct whether you are proving hardness or\nborrowing an algorithm. Master a few paradigms deeply and you can attack problems\nyou have never seen.",[381,5179,5180,5181,446,5196,5214],{},"If this material drew you in, the natural sequels are advanced algorithms and data\nstructures (Fibonacci heaps, the engineering behind Dijkstra\u002FPrim and disjoint-set\nunion), complexity and computability theory (the formal machinery beneath\n",[450,5182,5184],{"className":5183},[453],[450,5185,5187],{"className":5186,"ariaHidden":458},[457],[450,5188,5190,5193],{"className":5189},[462],[450,5191],{"className":5192,"style":467},[466],[450,5194,3342],{"className":5195},[471,475],[450,5197,5199],{"className":5198},[453],[450,5200,5202],{"className":5201,"ariaHidden":458},[457],[450,5203,5205,5208],{"className":5204},[462],[450,5206],{"className":5207,"style":467},[466],[450,5209,5211],{"className":5210},[471],[450,5212,476],{"className":5213},[471,475],"), and the specialized branches (randomized,\nstreaming, distributed, and geometric algorithms) where each of the paradigms\nabove reappears in a new guise.",[1070,5216,5218],{"id":5217},"takeaways","Takeaways",[390,5220,5221,5246,5428,5435,5444,5450],{},[393,5222,5223,5241,5242,5245],{},[450,5224,5226],{"className":5225},[453],[450,5227,5229],{"className":5228,"ariaHidden":458},[457],[450,5230,5232,5235],{"className":5231},[462],[450,5233],{"className":5234,"style":467},[466],[450,5236,5238],{"className":5237},[471],[450,5239,476],{"className":5240},[471,475],"-hardness rules out a fast exact algorithm in the ",[419,5243,5244],{},"worst case",",\nnot useful answers in practice. There are four honest responses.",[393,5247,5248,5249,5251,5252,5255,5256,5324,5325,5328,5329,5332,5333,416],{},"An ",[385,5250,1087],{}," trades optimality for a ",[419,5253,5254],{},"provable ratio"," ",[450,5257,5259],{"className":5258},[453],[450,5260,5262,5280],{"className":5261,"ariaHidden":458},[457],[450,5263,5265,5268,5271,5274,5277],{"className":5264},[462],[450,5266],{"className":5267,"style":1206},[466],[450,5269,1192],{"className":5270,"style":1191},[471,1114],[450,5272],{"className":5273,"style":1154},[1153],[450,5275,1216],{"className":5276},[1158],[450,5278],{"className":5279,"style":1154},[1153],[450,5281,5283,5286,5289,5292],{"className":5282},[462],[450,5284],{"className":5285,"style":1248},[466],[450,5287,1115],{"className":5288},[471,1114],[450,5290],{"className":5291,"style":1915},[1153],[450,5293,5295,5298],{"className":5294},[471],[450,5296,1192],{"className":5297,"style":1191},[471,1114],[450,5299,5301],{"className":5300},[1258],[450,5302,5304],{"className":5303},[1262],[450,5305,5307],{"className":5306},[1266],[450,5308,5310],{"className":5309,"style":1271},[1270],[450,5311,5312,5315],{"style":1274},[450,5313],{"className":5314,"style":1279},[1278],[450,5316,5318],{"className":5317},[1283,1284,1285,1286],[450,5319,5321],{"className":5320},[471,1286],[450,5322,1293],{"className":5323},[471,1286],". The proofs lean on a ",[385,5326,5327],{},"lower bound"," for the unknown optimum,\nas in the ",[385,5330,5331],{},"2-approximation for vertex cover",", which takes both endpoints of\na maximal matching, giving ",[450,5334,5336],{"className":5335},[453],[450,5337,5339,5363,5387],{"className":5338,"ariaHidden":458},[457],[450,5340,5342,5345,5348,5351,5354,5357,5360],{"className":5341},[462],[450,5343],{"className":5344,"style":1395},[466],[450,5346,2415],{"className":5347},[471],[450,5349,1192],{"className":5350,"style":1191},[471,1114],[450,5352,2415],{"className":5353},[471],[450,5355],{"className":5356,"style":1154},[1153],[450,5358,1460],{"className":5359},[1158],[450,5361],{"className":5362,"style":1154},[1153],[450,5364,5366,5369,5372,5375,5378,5381,5384],{"className":5365},[462],[450,5367],{"className":5368,"style":1395},[466],[450,5370,2440],{"className":5371},[471],[450,5373,2302],{"className":5374,"style":2301},[471,1114],[450,5376,2415],{"className":5377},[471],[450,5379],{"className":5380,"style":1154},[1153],[450,5382,1216],{"className":5383},[1158],[450,5385],{"className":5386,"style":1154},[1153],[450,5388,5390,5393,5396],{"className":5389},[462],[450,5391],{"className":5392,"style":1271},[466],[450,5394,1473],{"className":5395},[471],[450,5397,5399,5402],{"className":5398},[471],[450,5400,1192],{"className":5401,"style":1191},[471,1114],[450,5403,5405],{"className":5404},[1258],[450,5406,5408],{"className":5407},[1262],[450,5409,5411],{"className":5410},[1266],[450,5412,5414],{"className":5413,"style":1271},[1270],[450,5415,5416,5419],{"style":1274},[450,5417],{"className":5418,"style":1279},[1278],[450,5420,5422],{"className":5421},[1283,1284,1285,1286],[450,5423,5425],{"className":5424},[471,1286],[450,5426,1293],{"className":5427},[471,1286],[393,5429,5430,446,5432,5434],{},[385,5431,372],{},[385,5433,3804],{}," (2-opt, simulated annealing, tabu search)\nare fast and flexible but unguaranteed; validate them empirically.",[393,5436,5437,5440,5441,5443],{},[385,5438,5439],{},"Branch and bound"," finds the exact optimum by searching a tree and\n",[385,5442,4291],{}," subtrees whose optimistic bound cannot beat the incumbent; still\nexponential in the worst case, often fast in practice.",[393,5445,5446,5449],{},[385,5447,5448],{},"Special structure"," (trees, planar or bounded-treewidth graphs, small\nparameters, modest numeric values) frequently turns a worst-case-hard problem\ntractable on the instances you actually face.",[393,5451,5452,5455,5456,5474],{},[385,5453,5454],{},"Capstone view."," The course is a small kit of composable paradigms (divide\nand conquer, dynamic programming, greedy, graph methods, network flow) bounded\nby reductions and ",[450,5457,5459],{"className":5458},[453],[450,5460,5462],{"className":5461,"ariaHidden":458},[457],[450,5463,5465,5468],{"className":5464},[462],[450,5466],{"className":5467,"style":467},[466],[450,5469,5471],{"className":5470},[471],[450,5472,476],{"className":5473},[471,475],"-completeness. When hardness is real, you change\nthe question (approximation, heuristic, exact-but-exponential, special case)\nrather than abandon it.",[5476,5477,5480,5485],"section",{"className":5478,"dataFootnotes":376},[5479],"footnotes",[1070,5481,5484],{"className":5482,"id":2691},[5483],"sr-only","Footnotes",[5486,5487,5488,5614,5625,5637],"ol",{},[393,5489,5491,5494,5495,5510,5511,5606,5607],{"id":5490},"user-content-fn-clrs-vc",[385,5492,5493],{},"CLRS",", Ch. 35 — Approximation Algorithms (§35.1): the ",[450,5496,5498],{"className":5497},[453],[450,5499,5501],{"className":5500,"ariaHidden":458},[457],[450,5502,5504,5507],{"className":5503},[462],[450,5505],{"className":5506,"style":1169},[466],[450,5508,1473],{"className":5509},[471],"-approximation for vertex cover via a maximal matching lower bound, ",[450,5512,5514],{"className":5513},[453],[450,5515,5517,5541,5565],{"className":5516,"ariaHidden":458},[457],[450,5518,5520,5523,5526,5529,5532,5535,5538],{"className":5519},[462],[450,5521],{"className":5522,"style":1395},[466],[450,5524,2415],{"className":5525},[471],[450,5527,1192],{"className":5528,"style":1191},[471,1114],[450,5530,2415],{"className":5531},[471],[450,5533],{"className":5534,"style":1154},[1153],[450,5536,1460],{"className":5537},[1158],[450,5539],{"className":5540,"style":1154},[1153],[450,5542,5544,5547,5550,5553,5556,5559,5562],{"className":5543},[462],[450,5545],{"className":5546,"style":1395},[466],[450,5548,2440],{"className":5549},[471],[450,5551,2302],{"className":5552,"style":2301},[471,1114],[450,5554,2415],{"className":5555},[471],[450,5557],{"className":5558,"style":1154},[1153],[450,5560,1216],{"className":5561},[1158],[450,5563],{"className":5564,"style":1154},[1153],[450,5566,5568,5571,5574],{"className":5567},[462],[450,5569],{"className":5570,"style":1271},[466],[450,5572,1473],{"className":5573},[471],[450,5575,5577,5580],{"className":5576},[471],[450,5578,1192],{"className":5579,"style":1191},[471,1114],[450,5581,5583],{"className":5582},[1258],[450,5584,5586],{"className":5585},[1262],[450,5587,5589],{"className":5588},[1266],[450,5590,5592],{"className":5591,"style":1271},[1270],[450,5593,5594,5597],{"style":1274},[450,5595],{"className":5596,"style":1279},[1278],[450,5598,5600],{"className":5599},[1283,1284,1285,1286],[450,5601,5603],{"className":5602},[471,1286],[450,5604,1293],{"className":5605},[471,1286],". ",[406,5608,5613],{"href":5609,"ariaLabel":5610,"className":5611,"dataFootnoteBackref":376},"#user-content-fnref-clrs-vc","Back to reference 1",[5612],"data-footnote-backref","↩",[393,5615,5617,5619,5620],{"id":5616},"user-content-fn-clrs-ptas",[385,5618,5493],{},", Ch. 35 — Approximation Algorithms: polynomial-time approximation schemes (PTAS), including Knapsack and Euclidean TSP. ",[406,5621,5613],{"href":5622,"ariaLabel":5623,"className":5624,"dataFootnoteBackref":376},"#user-content-fnref-clrs-ptas","Back to reference 2",[5612],[393,5626,5628,5631,5632],{"id":5627},"user-content-fn-skiena-heur",[385,5629,5630],{},"Skiena",", §11 — NP-Completeness; Heuristics: local search and metaheuristics, and validating unguaranteed heuristics against known optima and lower bounds. ",[406,5633,5613],{"href":5634,"ariaLabel":5635,"className":5636,"dataFootnoteBackref":376},"#user-content-fnref-skiena-heur","Back to reference 3",[5612],[393,5638,5640,5643,5644,5715,5716],{"id":5639},"user-content-fn-erickson-struct",[385,5641,5642],{},"Erickson",", Ch. 12 — NP-Hardness: exploiting special structure, including fixed-parameter tractability such as ",[450,5645,5647],{"className":5646},[453],[450,5648,5650,5703],{"className":5649,"ariaHidden":458},[457],[450,5651,5653,5656,5659,5662,5694,5697,5700],{"className":5652},[462],[450,5654],{"className":5655,"style":4946},[466],[450,5657,4871],{"className":5658,"style":4870},[471,1114],[450,5660,2068],{"className":5661},[1875],[450,5663,5665,5668],{"className":5664},[471],[450,5666,1473],{"className":5667},[471],[450,5669,5671],{"className":5670},[1258],[450,5672,5674],{"className":5673},[1262],[450,5675,5677],{"className":5676},[1266],[450,5678,5680],{"className":5679,"style":4971},[1270],[450,5681,5682,5685],{"style":1274},[450,5683],{"className":5684,"style":1279},[1278],[450,5686,5688],{"className":5687},[1283,1284,1285,1286],[450,5689,5691],{"className":5690},[471,1286],[450,5692,4817],{"className":5693,"style":4816},[471,1114,1286],[450,5695],{"className":5696,"style":1233},[1153],[450,5698,1238],{"className":5699},[1237],[450,5701],{"className":5702,"style":1233},[1153],[450,5704,5706,5709,5712],{"className":5705},[462],[450,5707],{"className":5708,"style":1395},[466],[450,5710,4843],{"className":5711},[471,1114],[450,5713,2087],{"className":5714},[1957]," vertex cover. ",[406,5717,5613],{"href":5718,"ariaLabel":5719,"className":5720,"dataFootnoteBackref":376},"#user-content-fnref-erickson-struct","Back to reference 4",[5612],[5722,5723,5724],"style",{},"html .default .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .dark-mode .shiki span {color: var(--shiki-dark-mode);background: var(--shiki-dark-mode-bg);font-style: var(--shiki-dark-mode-font-style);font-weight: var(--shiki-dark-mode-font-weight);text-decoration: var(--shiki-dark-mode-text-decoration);}html.dark-mode .shiki span {color: var(--shiki-dark-mode);background: var(--shiki-dark-mode-bg);font-style: var(--shiki-dark-mode-font-style);font-weight: var(--shiki-dark-mode-font-weight);text-decoration: var(--shiki-dark-mode-text-decoration);}",{"title":376,"searchDepth":18,"depth":18,"links":5726},[5727,5730,5731,5732,5733,5734,5735,5736],{"id":1072,"depth":18,"text":1073,"children":5728},[5729],{"id":2026,"depth":24,"text":2027},{"id":3794,"depth":18,"text":3795},{"id":4280,"depth":18,"text":4281},{"id":4716,"depth":18,"text":4717},{"id":5073,"depth":18,"text":5074},{"id":5131,"depth":18,"text":5132},{"id":5217,"depth":18,"text":5218},{"id":2691,"depth":18,"text":5484},"This is the capstone of the course, so let us first take stock of the toolkit we\nhave assembled. Almost every algorithm we built fell under a handful of\nparadigms, recurring shapes of attack that transcend any single problem:","md",{"moduleNumber":196,"lessonNumber":24,"order":5740},803,true,[5743,5747,5750,5753],{"title":5744,"slug":5745,"difficulty":5746},"Combination Sum","combination-sum","Medium",{"title":5748,"slug":5749,"difficulty":5746},"Word Search","word-search",{"title":5751,"slug":5752,"difficulty":5746},"Matchsticks to Square","matchsticks-to-square",{"title":5754,"slug":5755,"difficulty":5756},"Maximum Score Words Formed by Letters","maximum-score-words-formed-by-letters","Hard","---\ntitle: Coping with NP-Hardness\nmodule: Intractability\nmoduleNumber: 8\nlessonNumber: 3\norder: 803\nsummary: >-\n  Proving a problem $\\mathsf{NP}$-hard is the beginning, not the end. The world\n  still needs answers. This lesson surveys the four honest responses to\n  hardness: approximation algorithms with a provable ratio (worked through a\n  2-approximation for vertex cover), heuristics and local search, exact\n  exponential methods like branch and bound, and exploiting special structure\n  in the instances you actually face.\ntopics: [Approximation, Heuristics]\nsources:\n  - book: CLRS\n    ref: \"Ch. 35 — Approximation Algorithms\"\n  - book: Skiena\n    ref: \"§11 — NP-Completeness; Heuristics\"\n  - book: Erickson\n    ref: \"Ch. 12 — NP-Hardness\"\npractice:\n  - title: 'Combination Sum'\n    slug: combination-sum\n    difficulty: Medium\n  - title: 'Word Search'\n    slug: word-search\n    difficulty: Medium\n  - title: 'Matchsticks to Square'\n    slug: matchsticks-to-square\n    difficulty: Medium\n  - title: 'Maximum Score Words Formed by Letters'\n    slug: maximum-score-words-formed-by-letters\n    difficulty: Hard\n---\n\nThis is the capstone of the course, so let us first take stock of the toolkit we\nhave assembled. Almost every algorithm we built fell under a handful of\n**paradigms**, recurring shapes of attack that transcend any single problem:\n\n- **Divide and conquer.** Split, recurse, combine; analyzed by recurrences and\n  the master theorem (merge sort, counting inversions, closest points, Karatsuba,\n  Strassen).\n- **Dynamic programming.** Tabulate overlapping subproblems (LCS, interval\n  scheduling, Bellman–Ford, Floyd–Warshall, [knapsack](\u002Falgorithms\u002Fdynamic-programming\u002Fknapsack), subset-sum).\n- **[Greedy](\u002Falgorithms\u002Fgreedy\u002Fthe-greedy-method).** Commit to a locally best choice and prove it is safe (Huffman,\n  Dijkstra, Prim, Kruskal; note that Dijkstra and Prim share the _same_ greedy\n  skeleton, differing only in the priority key).\n- **Graph methods.** BFS, DFS, topological sort, strongly connected components,\n  and the shortest-path and [spanning-tree](\u002Falgorithms\u002Fgraphs\u002Fminimum-spanning-trees) algorithms built atop them.\n- **Network flow.** Max-flow \u002F min-cut (Ford–Fulkerson, Edmonds–Karp) as a\n  modeling lens that absorbs matching, scheduling, and connectivity problems.\n- **[Reductions](\u002Falgorithms\u002Fintractability\u002Fp-np-reductions) and [$\\mathsf{NP}$-completeness](\u002Falgorithms\u002Fintractability\u002Fnp-completeness)**, the meta-tool: relate one\n  problem to another, transferring either an algorithm or a hardness proof.\n\nThat last paradigm is the hinge on which this lesson turns. Suppose you have\nfollowed the recipe from the previous lesson and proved that the problem on your\ndesk is $\\mathsf{NP}$-hard. This is genuine progress: you now know not to waste\nmonths hunting for a fast exact algorithm that almost certainly does not exist.\nBut the problem has not gone away. The trucks still need routing, the chips still\nneed placing, the schedule still must be made. $\\mathsf{NP}$-hardness is a\nstatement about the _worst case over all instances_; it does not forbid doing\nwell on the instances you actually meet, or doing _nearly_ as well as optimal, or\ndoing exactly as well but slowly.\n\nThere are four honest ways to cope, and a well-designed system often blends\nthem. We can **approximate**: settle for a solution provably close to optimal.\nWe can use **heuristics**, methods that work well in practice without\nguarantees. We can pay for an **exact exponential** algorithm that is merely\n_smart_ about its exponential search. Or we can exploit **special structure**\nin our instances. We take each in turn.\n\nThe decision tree below is the through-line of the whole subject in one picture:\n_first_ ask whether the problem is even hard; only the rightmost branch forces\nthe compromises of this lesson.\n\n$$\n% caption: Decision tree choosing a coping strategy for a possibly NP-hard problem.\n\\begin{tikzpicture}[\n    >=Stealth, node distance=9mm,\n    q\u002F.style={draw, rounded corners=1pt, minimum height=17mm, inner sep=4pt,\n              align=center, font=\\small, very thick},\n    a\u002F.style={draw, rounded corners=4pt, minimum height=17mm, inner sep=4pt,\n              align=center, font=\\small},\n    lbl\u002F.style={font=\\scriptsize}]\n  \\node (np) [q] {Is it\\\\$\\mathsf{NP}$-hard?};\n  \\node (poly) [a, left=18mm of np] {Use a known\\\\poly-time\\\\algorithm};\n  \\node (struct) [q, below=12mm of np] {Special\\\\structure?};\n  \\node (exploit) [a, right=16mm of struct] {Exploit it:\\\\tree \u002F planar,\\\\small $k$, pseudo-poly};\n  \\node (need) [q, below=12mm of struct] {Need a\\\\guarantee?};\n  \\node (approx) [a, left=16mm of need] {Approximation\\\\algorithm\\\\($\\rho$-ratio)};\n  \\node (exact) [q, below=12mm of need] {Exact\\\\optimum?};\n  \\node (bnb) [a, right=16mm of exact] {Branch \\&\\\\bound};\n  \\node (heur) [a, below=11mm of exact] {Heuristic \u002F\\\\local search};\n  \\draw[->] (np) -- node[lbl, above] {no} (poly);\n  \\draw[->] (np) -- node[lbl, right] {yes} (struct);\n  \\draw[->] (struct) -- node[lbl, above] {yes} (exploit);\n  \\draw[->] (struct) -- node[lbl, right] {no} (need);\n  \\draw[->] (need) -- node[lbl, above] {yes} (approx);\n  \\draw[->] (need) -- node[lbl, right] {no} (exact);\n  \\draw[->] (exact) -- node[lbl, above] {yes} (bnb);\n  \\draw[->] (exact) -- node[lbl, right] {no} (heur);\n\\end{tikzpicture}\n$$\n\n## Approximation algorithms\n\nThe most intellectually satisfying response is to relax _optimality_ while\nkeeping a _guarantee_. An **approximation algorithm** runs in polynomial time\nand returns a solution provably within a bounded factor of the best possible.\n\n> **Definition ($\\rho$-approximation).** For a minimization problem, an algorithm is a $\\rho$-approximation (with\n> $\\rho \\ge 1$) if, on every instance, it runs in polynomial time and returns a\n> solution of cost $C$ satisfying\n> $$ C \\le \\rho \\cdot C^{*}, $$\n> where $C^{*}$ is the cost of an optimal solution. The factor $\\rho$ is the\n> **approximation ratio**. (For a maximization problem the bound flips to $C\n> \\ge C^{*} \u002F \\rho$.)\n\nA ratio of $\\rho = 2$ means \"never more than twice optimal\": guaranteed, on\nevery input, with no exceptions. The subtlety, and the beauty, is that we prove\nthis _without ever knowing $C^{*}$_. The trick is almost always a **lower\nbound**: find some quantity that provably under-estimates $C^{*}$, then show\nyour solution is not much bigger than _that_.\n\n$$\n% caption: A $\\rho$-approximation pins the output cost $C$ into the band\n%          $[C^{*}, \\rho\\, C^{*}]$ — proved via a lower bound $L \\le C^{*}$.\n\\begin{tikzpicture}[>=Stealth, font=\\small]\n  \\definecolor{acc}{HTML}{2348F2}\n  \\draw[->, thick] (0,0) -- (11,0);\n  \\node[anchor=north] at (11,-0.1) {cost};\n  % lower bound L\n  \\node[circle, draw, fill=black!12, inner sep=1.6pt] (L) at (1.6,0) {};\n  \\node[anchor=south, font=\\scriptsize] at (1.6,0.15) {$L$ (lower bound)};\n  % optimum C*\n  \\node[circle, draw, fill=black!12, inner sep=1.9pt] (C) at (4.2,0) {};\n  \\node[anchor=south, font=\\scriptsize] at (4.2,0.15) {$C^{*}$ (optimum)};\n  % our output C\n  \\node[circle, draw=acc, fill=acc!15, very thick, inner sep=1.9pt] (Cout) at (7.6,0) {};\n  \\node[anchor=south, font=\\scriptsize, text=acc] at (7.6,0.15) {$C$ (output)};\n  % rho C*\n  \\node[circle, draw, inner sep=1.9pt] (R) at (9.4,0) {};\n  \\node[anchor=south, font=\\scriptsize] at (9.4,0.15) {$\\rho\\, C^{*}$};\n  % feasible band bracket under axis\n  \\draw[acc, thick] (4.2,-0.45) -- (9.4,-0.45);\n  \\draw[acc] (4.2,-0.35) -- (4.2,-0.55);\n  \\draw[acc] (9.4,-0.35) -- (9.4,-0.55);\n  \\node[anchor=north, font=\\scriptsize, text=acc] at (6.8,-0.55)\n    {guaranteed band: $C \\le \\rho\\, C^{*}$};\n  \\node[anchor=north, font=\\scriptsize] at (2.9,-0.55) {$L \\le C^{*}$ (unknown)};\n\\end{tikzpicture}\n$$\n\n### A worked example: 2-approximation for Vertex Cover\n\nA **vertex cover** of a graph $G = (V, E)$ is a set of vertices touching every\nedge: for each edge, at least one endpoint is chosen. $\\textsc{Min-Vertex-Cover}$, the\nproblem of finding the smallest such set, is $\\mathsf{NP}$-hard. Yet a\ndisarmingly simple algorithm comes within a factor of $2$.\n\n```algorithm\ncaption: $\\textsc{Approx-Vertex-Cover}(G)$ — return a cover at most twice optimal\nnumber: 1\n$C \\gets \\emptyset$\n$E' \\gets E$ \u002F\u002F working copy\nwhile $E' \\neq \\emptyset$ do\n  pick any edge $(u, v) \\in E'$\n  $C \\gets C \\cup \\{u, v\\}$ \u002F\u002F take both endpoints\n  remove from $E'$ every edge incident to $u$ or $v$\nreturn $C$\n```\n\nThe algorithm repeatedly grabs an uncovered edge and throws _both_ its\nendpoints into the cover. Taking both feels wasteful (surely one would do?),\nbut it is exactly what makes the analysis work.\n\n**Correctness.** The loop continues until $E'$ is empty, i.e. until every edge\nis incident to some chosen vertex. So $C$ is a valid vertex cover.\n\n**The ratio.** Let $M$ be the set of edges _picked_ in the loop (one per\niteration). No two edges in $M$ share an endpoint: once we pick $(u,v)$, we\ndelete every edge touching $u$ or $v$, so no later picked edge can reuse them.\nA set of edges sharing no endpoints is a **matching**. Now the two key facts:\n\n- Our cover has $|C| = 2|M|$, two fresh vertices per picked edge.\n- _Any_ vertex cover, including the optimal one, must contain at least one\n  endpoint of each edge in $M$; since those endpoints are all distinct, $C^{*}\n  \\ge |M|$.\n\nChaining these,\n$$ |C| = 2|M| \\le 2\\,C^{*}, $$\nso $\\textsc{Approx-Vertex-Cover}$ is a $2$-approximation.[^clrs-vc] We bounded our output\nagainst $|M|$, a quantity that lower-bounds the unknown optimum, the\nsignature move of approximation analysis. (Whether vertex cover admits a ratio\nbetter than $2$ is a famous open question; under standard hardness\nassumptions, no polynomial-time algorithm does substantially better.)\n\n$$\n% caption: $\\textsc{Approx-Vertex-Cover}$: each picked matching edge (heavy) contributes\n%          both endpoints (filled) to the cover, giving $|C| = 2|M| \\le 2C^{*}$.\n\\begin{tikzpicture}[>=Stealth, font=\\small,\n    v\u002F.style={circle, draw, inner sep=1.7pt, minimum size=5mm},\n    cov\u002F.style={circle, draw=acc, fill=acc!15, very thick, inner sep=1.7pt, minimum size=5mm}]\n  \\definecolor{acc}{HTML}{2348F2}\n  \\node[cov] (a) at (0,1.4) {$a$};\n  \\node[cov] (b) at (2,1.4) {$b$};\n  \\node[cov] (c) at (4,1.4) {$c$};\n  \\node[cov] (d) at (6,1.4) {$d$};\n  \\node[v]   (e) at (1,0) {$e$};\n  \\node[v]   (f) at (3,0) {$f$};\n  \\node[v]   (g) at (5,0) {$g$};\n  % matching edges (heavy, accent)\n  \\draw[acc, very thick] (a)--(b);\n  \\draw[acc, very thick] (c)--(d);\n  % other edges covered by the chosen endpoints\n  \\draw (a)--(e);\n  \\draw (b)--(f);\n  \\draw (c)--(f);\n  \\draw (c)--(g);\n  \\draw (d)--(g);\n  \\node[anchor=south, font=\\scriptsize, text=acc, fill=white, inner sep=1.5pt] at (1,1.62) {$M$: edge 1};\n  \\node[anchor=south, font=\\scriptsize, text=acc, fill=white, inner sep=1.5pt] at (5,1.62) {$M$: edge 2};\n  \\node[anchor=north, align=center, font=\\scriptsize] at (3,-0.4)\n    {cover $C=\\{a,b,c,d\\}$, $|C|=2|M|=4$; every edge touches a filled vertex};\n\\end{tikzpicture}\n$$\n\nNot every problem is so cooperative. Approximability varies wildly:\n\n- Some problems admit a **polynomial-time approximation scheme** (PTAS): for any\n  $\\varepsilon > 0$ a $(1+\\varepsilon)$-approximation in polynomial time: you\n  can get as close to optimal as you like, paying in running time.\n  **Euclidean TSP** and **Knapsack** are of this kind.[^clrs-ptas]\n- Some have a fixed best-possible constant ratio, like vertex cover's $2$.\n- Some are **inapproximable**: unless $\\mathsf{P} = \\mathsf{NP}$, no\n  polynomial-time algorithm achieves _any_ constant ratio. **General TSP**\n  (without triangle inequality) is the classic example: even approximating it\n  within any factor is $\\mathsf{NP}$-hard.\n\nWhen the triangle inequality _does_ hold (the **metric** case), a clean\n$2$-approximation falls out of the [minimum spanning tree](\u002Falgorithms\u002Fgraphs\u002Fminimum-spanning-trees): double every MST\nedge to get an Eulerian multigraph, walk an Euler tour, and **shortcut** past\nalready-visited vertices. The shortcuts only shorten the walk (triangle\ninequality), so the tour costs at most $2\\,\\text{MST} \\le 2\\,\\text{OPT}$, since\nthe MST is no heavier than the optimal tour with one edge removed.\n\n$$\n% caption: Metric-TSP $2$-approximation: double the MST into an Euler tour, then shortcut\n%          repeats; cost $\\le 2\\,\\text{MST} \\le 2\\,\\text{OPT}$.\n\\begin{tikzpicture}[>=Stealth, font=\\small,\n    v\u002F.style={circle, draw, fill=black!6, inner sep=1.7pt, minimum size=5mm}]\n  \\definecolor{acc}{HTML}{2348F2}\n  % left: MST\n  \\begin{scope}\n    \\node[v] (a) at (0,2) {$a$};\n    \\node[v] (b) at (1.6,2.4) {$b$};\n    \\node[v] (c) at (1.2,0.7) {$c$};\n    \\node[v] (d) at (2.8,1.2) {$d$};\n    \\node[v] (e) at (0.2,0.5) {$e$};\n    \\draw[acc, very thick] (a)--(b);\n    \\draw[acc, very thick] (a)--(c);\n    \\draw[acc, very thick] (c)--(d);\n    \\draw[acc, very thick] (a)--(e);\n    \\node[anchor=north, font=\\scriptsize] at (1.3,-0.1) {MST $T$ (cost $\\le$ OPT)};\n  \\end{scope}\n  % arrow\n  \\draw[->, thick] (3.4,1.3) -- node[above, font=\\scriptsize]{double +} node[below, font=\\scriptsize]{shortcut} (5.0,1.3);\n  % right: shortcut tour\n  \\begin{scope}[shift={(5.6,0)}]\n    \\node[v] (a) at (0,2) {$a$};\n    \\node[v] (b) at (1.6,2.4) {$b$};\n    \\node[v] (c) at (1.2,0.7) {$c$};\n    \\node[v] (d) at (2.8,1.2) {$d$};\n    \\node[v] (e) at (0.2,0.5) {$e$};\n    % a Hamiltonian cycle a-b-d-c-e-a\n    \\draw[acc, very thick] (a)--(b);\n    \\draw[acc, very thick] (b)--(d);\n    \\draw[acc, very thick] (d)--(c);\n    \\draw[acc, very thick] (c)--(e);\n    \\draw[acc, very thick] (e)--(a);\n    \\node[anchor=north, font=\\scriptsize] at (1.3,-0.1) {tour (cost $\\le 2\\,$OPT)};\n  \\end{scope}\n\\end{tikzpicture}\n$$\n\n## Heuristics and local search\n\nWhen a provable ratio is out of reach or simply unnecessary, we turn to\n**heuristics**, strategies that are usually good but carry no worst-case\npromise. The most general is **local search**: start from some feasible\nsolution and repeatedly apply a small modification, a _move_, that improves\nthe objective, stopping at a **local optimum** where no single move helps.\n\nFor the traveling salesman, the famous **2-opt** move deletes two edges of the\ncurrent tour and reconnects the pieces the other way; iterating it untangles\ncrossings and converges to short, though not always optimal, tours.\n\n$$\n% caption: A 2-opt move deletes two crossing edges (discarded, red) and reconnects the\n%          four endpoints the other way, uncrossing the tour and shortening it.\n\\begin{tikzpicture}[>=Stealth, font=\\small,\n    v\u002F.style={circle, draw, fill=black!6, inner sep=1.6pt, minimum size=4.5mm}]\n  \\definecolor{acc}{HTML}{2348F2}\n  % left: crossing tour\n  \\begin{scope}\n    \\node[v] (a) at (0,2)   {$a$};\n    \\node[v] (b) at (2.2,2) {$b$};\n    \\node[v] (c) at (0,0)   {$c$};\n    \\node[v] (d) at (2.2,0) {$d$};\n    % surrounding tour edges (kept)\n    \\draw[acc, very thick] (a) to[bend left=35] (b);\n    \\draw[acc, very thick] (c) to[bend right=35] (d);\n    % the two edges that cross (to be deleted)\n    \\draw[red!75!black, very thick] (a) -- (d);\n    \\draw[red!75!black, very thick] (c) -- (b);\n    \\node[anchor=north, font=\\scriptsize, text=red!75!black] at (1.1,-0.35)\n      {delete the two crossing edges};\n  \\end{scope}\n  \\draw[->, thick] (3.1,1.0) -- node[above, font=\\scriptsize]{2-opt} (4.5,1.0);\n  % right: uncrossed tour\n  \\begin{scope}[shift={(5.4,0)}]\n    \\node[v] (a) at (0,2)   {$a$};\n    \\node[v] (b) at (2.2,2) {$b$};\n    \\node[v] (c) at (0,0)   {$c$};\n    \\node[v] (d) at (2.2,0) {$d$};\n    \\draw[acc, very thick] (a) to[bend left=35] (b);\n    \\draw[acc, very thick] (c) to[bend right=35] (d);\n    % reconnected the other way (no crossing)\n    \\draw[acc, very thick] (a) -- (c);\n    \\draw[acc, very thick] (b) -- (d);\n    \\node[anchor=north, font=\\scriptsize, text=acc] at (1.1,-0.35)\n      {reconnect the other way: shorter};\n  \\end{scope}\n\\end{tikzpicture}\n$$\n\nLocal\nsearch has a characteristic failure mode: it can stall in a local optimum far\nfrom the global one. The standard escapes are _metaheuristics_ that\noccasionally accept worsening moves to climb out of bad valleys:\n\n- **Simulated annealing** accepts uphill moves with a probability that cools\n  over time, mimicking the physics of slowly freezing metal.\n- **Tabu search** forbids recently-visited solutions to avoid cycling back.\n- **Genetic algorithms** evolve a population of solutions by recombination and\n  mutation.\n\n$$\n% caption: Local search descends to the nearest valley and stalls at a local optimum; a\n%          metaheuristic must accept an uphill move to escape and reach the global\n%          optimum.\n\\begin{tikzpicture}[>=Stealth, font=\\small]\n  \\definecolor{acc}{HTML}{2348F2}\n  % axes\n  \\draw[->] (0,0) -- (0,2.9) node[anchor=south east, font=\\scriptsize] {cost};\n  \\draw[->] (0,0) -- (9.3,0) node[anchor=north east, font=\\scriptsize] {solution space};\n  % cost landscape with two controlled wells: shallow local well at x=2, deep global well at x=6.6\n  \\draw[thick]\n    (0.5,2.3)\n      .. controls (1.3,2.3) and (1.5,0.95) .. (2.0,0.95)   % into shallow local minimum\n      .. controls (2.5,0.95) and (2.9,2.25) .. (3.7,2.25)  % up over the ridge\n      .. controls (4.7,2.25) and (5.6,0.30) .. (6.6,0.30)  % down into deep global minimum\n      .. controls (7.4,0.30) and (7.8,1.6) .. (8.6,1.7);   % back up on the right\n  % local optimum marker, sitting in the shallow well\n  \\node[circle, draw, fill=black!12, inner sep=1.6pt] at (2.0,0.95) {};\n  \\node[anchor=south, font=\\scriptsize, align=center] at (2.0,1.18)\n    {local optimum\\\\(search stalls here)};\n  % global optimum marker, in the deep well\n  \\node[circle, draw=acc, fill=acc!15, very thick, inner sep=1.7pt] at (6.6,0.30) {};\n  \\node[anchor=west, font=\\scriptsize, text=acc, align=left] at (6.95,0.55)\n    {global\\\\optimum};\n  % escape arrow: uphill over the ridge toward the deeper well\n  \\draw[->, red!75!black, thick] (2.35,1.05) to[bend left=20] (4.05,1.55);\n  \\node[font=\\scriptsize, text=red!75!black, align=center, anchor=south] at (3.5,1.75)\n    {accept an uphill\\\\move to escape};\n\\end{tikzpicture}\n$$\n\nHeuristics are the workhorses of industry precisely because they are flexible\nand fast. Their cost is the loss of guarantees: you rarely know how far from\noptimal you landed. The honest practice, Skiena stresses, is to test against\nknown optima on small instances and against lower bounds on large ones.[^skiena-heur]\n\n## Exact exponential methods: branch and bound\n\nSometimes you genuinely need the _optimal_ answer and the instances are small\nenough to afford exponential time, provided it is spent wisely. The key idea is to\nsearch the space of solutions as a tree while **pruning** subtrees that\nprovably cannot beat the best solution found so far. This is **[branch and\nbound](\u002Falgorithms\u002Fbacktracking\u002Fbranch-and-bound)**.\n\nThe method interleaves two operations. _Branching_ splits the problem into\nsubproblems (e.g. \"vertex $v$ is in the cover\" vs. \"$v$ is out\"), forming a\nsearch tree. _Bounding_ computes, for each subproblem, an optimistic estimate,\na **bound**, on the best solution reachable within it. If that optimistic\nestimate is already no better than the best complete solution we have already\nfound (the **incumbent**), the entire subtree is discarded unexplored.\n\n> **Remark (The pruning rule).** For a minimization problem, maintain the incumbent cost\n> $U$ (best solution so far). At a subproblem with optimistic lower bound $L$,\n> if $L \\ge U$, _prune_ — nothing in this subtree can improve on the incumbent.\n\nThe worst case is still exponential; branch and bound does not repeal\n$\\mathsf{NP}$-hardness. But on real instances a good bound can prune away the\noverwhelming majority of the tree, making problems with thousands of variables\nroutinely solvable. The quality of the bound is everything: a tight bound (often\nfrom a relaxation such as linear programming) prunes aggressively; a loose one\nleaves a near-complete exponential search. Modern integer-programming solvers\nare, at heart, exquisitely engineered branch-and-bound engines.\n\n$$\n% caption: Branch and bound: split on a variable, bound each subtree, and prune any whose\n%          optimistic bound $L$ is no better than the incumbent $U$.\n\\begin{tikzpicture}[>=Stealth, font=\\small,\n    n\u002F.style={circle, draw, minimum size=8mm, inner sep=1pt},\n    pruned\u002F.style={circle, draw=red!75!black, dashed, minimum size=8mm, inner sep=1pt, fill=red!18},\n    keep\u002F.style={circle, draw=acc, fill=acc!15, very thick, minimum size=8mm, inner sep=1pt}]\n  \\definecolor{acc}{HTML}{2348F2}\n  \\node[n] (r) at (0,2.6) {};\n  \\node[n] (l) at (-2.4,1.2) {};\n  \\node[n] (rr) at (2.4,1.2) {};\n  \\node[keep] (ll) at (-3.6,-0.2) {};\n  \\node[pruned] (lr) at (-1.2,-0.2) {};\n  \\node[n] (rl) at (1.2,-0.2) {};\n  \\node[pruned] (rrr) at (3.6,-0.2) {};\n  \\draw[->] (r) -- node[above left, font=\\scriptsize]{$v=0$} (l);\n  \\draw[->] (r) -- node[above right, font=\\scriptsize]{$v=1$} (rr);\n  \\draw[->] (l) -- (ll);\n  \\draw[->] (l) -- (lr);\n  \\draw[->] (rr) -- (rl);\n  \\draw[->] (rr) -- (rrr);\n  \\node[font=\\scriptsize, text=acc, anchor=north] at (-3.6,-0.55) {incumbent $U$};\n  \\node[font=\\scriptsize, text=red!75!black, anchor=north, align=center] at (-1.2,-0.55) {$L\\ge U$\\\\prune};\n  \\node[font=\\scriptsize, text=red!75!black, anchor=north, align=center] at (3.6,-0.55) {$L\\ge U$\\\\prune};\n  \\node[font=\\scriptsize, anchor=west] at (0.3,2.6) {root: all solutions};\n\\end{tikzpicture}\n$$\n\n## Exploiting special structure\n\nThe final and most underrated strategy is to remember that\n$\\mathsf{NP}$-hardness is a worst case over _all_ inputs, and your inputs may\nnot be the worst. Many hard problems become _easy_ when restricted to the\nstructured instances that arise in practice.\n\n- **Restricted graph classes.** Problems that are $\\mathsf{NP}$-hard on general\n  graphs frequently admit polynomial (even linear) algorithms on **trees**,\n  on **bipartite** graphs, or on **planar** graphs. Independent Set, hard in\n  general, falls to a simple greedy\u002Fdynamic program on trees.\n- **Bounded parameters.** A problem may be solvable in time\n  $f(k) \\cdot n^{O(1)}$, where $k$ is some small parameter of the instance (the\n  solution size, the treewidth, the number of constraints). The exponential\n  blow-up is confined to $k$, so if $k$ is small the algorithm is fast. This is\n  the domain of **fixed-parameter tractability**: Vertex Cover, for instance, is\n  solvable in $O(2^{k} \\cdot n)$ for covers of size $k$, practical whenever the\n  cover is small even if the graph is huge.[^erickson-struct]\n- **Pseudo-polynomial algorithms.** Numeric problems like $\\textsc{Subset-Sum}$ and\n  **Knapsack** have dynamic programs running in time polynomial in the\n  _numeric values_, fast when the numbers are modest, exponential only because\n  values can be exponentially large in their bit-length.\n\nThe lesson is to look hard at the instances you must actually solve before\ndeclaring defeat. Hardness in the worst case is fully compatible with easiness\nin your case.\n\n## Choosing a strategy\n\nThese responses are not rivals so much as a toolkit; the right choice depends on\nwhat you can tolerate.\n\n- Need a _guarantee_ and can accept \"near-optimal\"? Reach for an\n  **approximation algorithm**.\n- Need _speed_ and _flexibility_ and can live without guarantees? Use a\n  **heuristic** or **local search**, validated empirically.\n- Need the _exact_ optimum on instances of modest size? Invest in **branch and\n  bound** with the tightest bound you can compute.\n- Do your real instances have _structure_, such as small parameters, special graph\n  shape, or modest numbers? **Exploit it**, possibly turning the problem\n  polynomial outright.\n\n## Final thoughts\n\nStep back and the whole course resolves into a single arc. We learned to _analyze_\n(asymptotics, recurrences, invariants), to _design_ across a small repertoire of\nparadigms (divide and conquer, dynamic programming, greedy, graph search, network\nflow), and finally to _recognize the limits of design_ through reductions and\n$\\mathsf{NP}$-completeness. This last lesson closes the loop: when the limit is\nreal, you do not give up — you change the question, trading exactness, optimality,\nor generality for tractability, and doing so _honestly_, with a clear statement of\nwhat you gave up.\n\nThe paradigms compose. A branch-and-bound solver leans on a relaxation (often\nlinear programming) for its bound; an approximation proof leans on a combinatorial\nlower bound like a matching; a special-case algorithm is frequently just dynamic\nprogramming rediscovered on a tree. The reduction habit — _map my problem onto one\nI already understand_ — is the same instinct whether you are proving hardness or\nborrowing an algorithm. Master a few paradigms deeply and you can attack problems\nyou have never seen.\n\nIf this material drew you in, the natural sequels are advanced algorithms and data\nstructures (Fibonacci heaps, the engineering behind Dijkstra\u002FPrim and disjoint-set\nunion), complexity and computability theory (the formal machinery beneath\n$\\mathsf{P}$ and $\\mathsf{NP}$), and the specialized branches (randomized,\nstreaming, distributed, and geometric algorithms) where each of the paradigms\nabove reappears in a new guise.\n\n## Takeaways\n\n- $\\mathsf{NP}$-hardness rules out a fast exact algorithm in the _worst case_,\n  not useful answers in practice. There are four honest responses.\n- An **approximation algorithm** trades optimality for a _provable ratio_ $C \\le\n  \\rho\\, C^{*}$. The proofs lean on a **lower bound** for the unknown optimum,\n  as in the **2-approximation for vertex cover**, which takes both endpoints of\n  a maximal matching, giving $|C| = 2|M| \\le 2C^{*}$.\n- **Heuristics** and **local search** (2-opt, simulated annealing, tabu search)\n  are fast and flexible but unguaranteed; validate them empirically.\n- **Branch and bound** finds the exact optimum by searching a tree and\n  **pruning** subtrees whose optimistic bound cannot beat the incumbent; still\n  exponential in the worst case, often fast in practice.\n- **Special structure** (trees, planar or bounded-treewidth graphs, small\n  parameters, modest numeric values) frequently turns a worst-case-hard problem\n  tractable on the instances you actually face.\n- **Capstone view.** The course is a small kit of composable paradigms (divide\n  and conquer, dynamic programming, greedy, graph methods, network flow) bounded\n  by reductions and $\\mathsf{NP}$-completeness. When hardness is real, you change\n  the question (approximation, heuristic, exact-but-exponential, special case)\n  rather than abandon it.\n\n[^clrs-vc]: **CLRS**, Ch. 35 — Approximation Algorithms (§35.1): the $2$-approximation for vertex cover via a maximal matching lower bound, $|C| = 2|M| \\le 2C^{*}$.\n[^clrs-ptas]: **CLRS**, Ch. 35 — Approximation Algorithms: polynomial-time approximation schemes (PTAS), including Knapsack and Euclidean TSP.\n[^skiena-heur]: **Skiena**, §11 — NP-Completeness; Heuristics: local search and metaheuristics, and validating unguaranteed heuristics against known optima and lower bounds.\n[^erickson-struct]: **Erickson**, Ch. 12 — NP-Hardness: exploiting special structure, including fixed-parameter tractability such as $O(2^{k}\\cdot n)$ vertex cover.\n",{"text":5759,"minutes":5760,"time":5761,"words":5762},"11 min read",10.71,642600,2142,{"title":368,"description":5737},[5765,5767,5769],{"book":5493,"ref":5766},"Ch. 35 — Approximation 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