Page 1 Lecture 26 181 Page 2 Lecture 26 181 193 Coping with NP-Completeness Is your real problem a special subcase? E.g. 3-SAT is NP-complete, but 2-SAT is not; ditto 3- vs 2- coloring E.g. you only need planar graphs, or degree 3 graphs, …? Guaranteed approximation good enough? E.g. Euclidean TSP within 2 * Opt in poly time Fast enough in practice (esp. if n is small), E.g. clever exhaustive search like backtrack, branch & bound, pruning Heuristics – usually a good approximation and/or usually fast Page 3 Lecture 26 181 193 Coping with NP-Completeness Is your real problem a special subcase? E.g. 3-SAT is NP-complete, but 2-SAT is not; ditto 3- vs 2- coloring E.g. you only need planar graphs, or degree 3 graphs, …? Guaranteed approximation good enough? E.g. Euclidean TSP within 2 * Opt in poly time Fast enough in practice (esp. if n is small), E.g. clever exhaustive search like backtrack, branch & bound, pruning Heuristics – usually a good approximation and/or usually fast 194 5 3 4 6 4 7 2 5 8 Example: b = 34 NP-complete problem: TSP Input: An undirected graph G=(V,E) with integer edge weights, and an integer b. Output: YES iff there is a simple cycle in G passing through all vertices (once), with total cost = b. Page 4 Lecture 26 181 193 Coping with NP-Completeness Is your real problem a special subcase? E.g. 3-SAT is NP-complete, but 2-SAT is not; ditto 3- vs 2- coloring E.g. you only need planar graphs, or degree 3 graphs, …? Guaranteed approximation good enough? E.g. Euclidean TSP within 2 * Opt in poly time Fast enough in practice (esp. if n is small), E.g. clever exhaustive search like backtrack, branch & bound, pruning Heuristics – usually a good approximation and/or usually fast 194 5 3 4 6 4 7 2 5 8 Example: b = 34 NP-complete problem: TSP Input: An undirected graph G=(V,E) with integer edge weights, and an integer b. Output: YES iff there is a simple cycle in G passing through all vertices (once), with total cost = b. 195 € lim n?8 NN OPT ?8 TSP - Nearest Neighbor Heuristic NN Heuristic –go to nearest unvisited vertex Fact: NN tour can be about (log n) x opt, i.e. (above example is not that bad) Page 5 Lecture 26 181 193 Coping with NP-Completeness Is your real problem a special subcase? E.g. 3-SAT is NP-complete, but 2-SAT is not; ditto 3- vs 2- coloring E.g. you only need planar graphs, or degree 3 graphs, …? Guaranteed approximation good enough? E.g. Euclidean TSP within 2 * Opt in poly time Fast enough in practice (esp. if n is small), E.g. clever exhaustive search like backtrack, branch & bound, pruning Heuristics – usually a good approximation and/or usually fast 194 5 3 4 6 4 7 2 5 8 Example: b = 34 NP-complete problem: TSP Input: An undirected graph G=(V,E) with integer edge weights, and an integer b. Output: YES iff there is a simple cycle in G passing through all vertices (once), with total cost = b. 195 € lim n?8 NN OPT ?8 TSP - Nearest Neighbor Heuristic NN Heuristic –go to nearest unvisited vertex Fact: NN tour can be about (log n) x opt, i.e. (above example is not that bad) 198 2x Approximation to EuclideanTSP A TSP tour visits all vertices, so contains a spanning tree, so TSP cost is > cost of min spanning tree. Find MST Find “DFS” Tour Shortcut TSP = shortcut < DFST = 2 * MST < 2 * TSP 5 3 4 2 5 6 4 7 8Read More

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