On the complexity of local search

We prove a number of complexity results on the computational paradigm of local op-timality. Our main results are these: (a) Finding a local optimum under the Lin-Kernighan heuristic for the traveling salesman problemis PLS-complete. (b) Finding stable configurations in neural networks in the Hopfield mode/is PLS-complete. (c) We show that a host of simple unweighted local optimality problems are P-complete. (d) We introduce a general framework for establishing exponential worst-case bounds for local optimization heuristics. (e) And we show that local search problems become PSPACE-complete if we insist that the local optimum returned be attainable by local improvements from a given initial solution.

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