Local search for intractable problems ps98 chapt 19
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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)

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Local search for intractable problems (PS98, chapt. 19)

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Local search for intractable problems ps98 chapt 19

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


Exact neighborhoods

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)


The tsp

The TSP


2 opt and 3 opt

2-opt and 3-opt


Local search for intractable problems ps98 chapt 19

Link emphasizing proven optimality: GA 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


Heuristics choices tradeoffs

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)

  • Keep dictionary of previous local optima to save time in checking final local optimality


Local search for intractable problems ps98 chapt 19

Min-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 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 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.)


Local search for intractable problems ps98 chapt 19

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?

  • Compare “linear 2-opt” with Concorde on big problems

  • Try “linear 2-opt” on some images?

  • Code and test “linear 3-opt”

  • Combine variable-depth and linear 2,3-opt?


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