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Local search for intractable problems (PS98, chapt. 19) Idea: for a feasible solution, define a neighborhood of feasible solutions Search neighborhood for a solution of lower cost; move to a better one (best or first-found) When current solution is locally optimal (nothing better in its neighborhood), stop. Repeat from many random initial feasible solutions

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Exact neighborhoods If locally optimal globally optimal, neighborhood is called “exact’’. Examples: Linear programming; Minimum spanning tree Having an exact neighborhood is a hint (but doesn’t prove) that a problem is “easy” (in P)

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The TSP

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2-opt and 3-opt

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Link emphasizing proven optimality: GA Tech pageGA Tech page important contributions of Shen Lin (1965): completely random starts, prob. of opt example, 48 cities: prob. 5%; with 100 runs, prob. opt = 1 -.95^100 = 99.4% strong neighborhoods work well with completely random starts weak neighborhoods are helped by good starts another contribution of Shen Lin: 3-opt much better than 2-opt; but 4-opt not that much better than 3-opt Lin's results on TSP problems were surprisingly good, and that led others to apply local search to other problems

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Heuristics, choices, tradeoffs First-improvement vs. steepest-descent Randomize search order? (May be useful if starting feasible solutions are scarce) Further improve local optima? Reduction (S. Lin): keep pieces representing common features of local optima Or, forbid these features in looking for new local optima (“denial” in SW68)SW68 Keep dictionary of previous local optima to save time in checking final local optimality

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Min-cost survivable networksMin-cost survivable networks (SWK 1969): find graph with given vertex-connectivity and min weight. X- change. Features: Starts and keeping feasible are key problems. Offshore natural-gas pipelinesOffshore natural-gas pipelines (RFSSK70): find min cost delivery system for offshore natural gas. Features: Costing is complicated and therefore expensive; Delta-change is a very small neighborhood. Uniform graph partitioningUniform graph partitioning (KL70): Split 2n nodes into two circuit boards so cost of inter-board edges is min. Stab for favorable sequence, accepting some down-turns (“variable-depth search”) until net is negative. (Applied to TSP in LK73.)

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Project suggestions: Visualize dynamics of 2-opt, 3-opt Apply local search to a (possibly new) combinatorial optimization problem: batting order? Exam scheduling? Drawing graphs with small number of crossovers? Untying knots? Try instances of an undecidable problem like Post Correspondence Problem? Convert half-tone pictures to tours Mona Lisa 100K problem using “linear 2-opt” [SW70], say?Mona Lisa 100K problem Compare “linear 2-opt” with Concorde on big problemslinear 2-optConcorde Try “linear 2-opt” on some images?linear 2-opt Code and test “linear 3-opt”linear 3-opt Combine variable-depth and linear 2,3-opt?

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