An Overview of What We Can and Cannot Do with Local Search
Résumé
Since we do not know any algorithm to efficiently solve the NP-hard optimization problems, a lot of approximation algorithms have been evolved. A general method for this purpose is Local Search. One assumes a neighboring structure between the solutions of an optimization problem and wants to find a solution that is the best in its neighborhood, called a local optimum, instead of the best solution in the domain. So, the standard local search heuristic starts from an initial solution and keeps moving to some better neighbor until it reaches a local optimum. This simple method turns out to be very successful in practice both in its running time performance and on the quality of the solutions that produces. The main purpose of this work is to sum up the theoretical work that has been done concerning the limits of local search in general and its proven approximation efficiency for particular problems. Hence, on the one hand we present the PLScompleteness theory and show that for the PLS-complete problems the standard local search heuristic takes exponential time in the worst case. We also show that if it is NP-hard to ε-approximate an optimization problem then there is no neighborhood which produces local optima only ε-close to global optima, unless NP=co-NP. On the other hand, we present numerous of examples of NP-hard optimization problems that under appropriate neighborhoods guarantee local optima ε-close to the global optima. Such guarantees are, in many cases, between the best ones for these problems by any algorithm. Local search heuristic is pseudopolynomial, so when the problems are unweighted or with polynomially bounded weights it reaches a local optimum in polynomial time.
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