Simple local search problems that are hard to solve

Reviewer: Vijay Varadachari Raghavan

The study of the complexity of finding locally optimal solutions to NP-hard search problems has led to the definition of a new class of problems by Johnson et al. [1]. This class, called <__?__Pub Fmt italic>polynomial local search<__?__Pub Fmt /italic> (PLS), contains the class <__?__Pub Fmt italic>P s <__?__Pub Fmt /italic> (the class <__?__Pub Fmt italic>P<__?__Pub Fmt /italic> extended to include general search problems). An important part of the specification of a search problem is the specification of the neighborhood of a solution. For a problem to be in PLS, polynomial-time algorithms must exist for the following three problems: computing an initial feasible solution, computing the cost of a solution, and selecting a better solution (if one exists) from the neighborhood of the current solution. In this paper, the authors show that several problems are PLS-complete (see Johnson et al. [1] for a definition). These problems include search problems such as MAX CUT, 2SAT, graph partitioning under single swap, and single swap based on the Fiduccia-Ma<__?__Pub Caret>ttheyses heuristic [2], as well as the finding of a stable configuration in a connectionist network. Whereas the above problems are PLS-complete, the authors prove that the corresponding unweighted problems (weights on edges are ?1 for stable configuration) are P-complete. This result is worth noting in light of the fact that P?P s ?PLS?NP s (<__?__Pub Fmt italic>NP s <__?__Pub Fmt /italic> is the nondeterministic counterpart of <__?__Pub Fmt italic>P s <__?__Pub Fmt /italic>). They also report that the traveling salesman problem under the Lin-Kernighan neighborhood [3] is PLS-complete. Although problems such as MAX CUT and graph partitioning had previously been shown to be PLS-complete [2], the completeness results in this paper are based on a much simpler neighborhood structure and therefore involve a smaller local search space. In other words, this paper shows that the simpler versions of these problems, which may appear to be quite tractable, are nevertheless PLS-complete.