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

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)
- When current solution is locally optimal (nothing better in its neighborhood), stop.
- Repeat from many random initial feasible solutions

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)

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

- 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

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.)

- 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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