Finding One Local Optimum Is Easy -- but What About Two?

TL;DR

This paper investigates the complexity of finding multiple locally optimal solutions in unweighted local search problems, revealing NP-hardness for computing two solutions in several classic problems, contrasting the single-solution case.

Contribution

It demonstrates that computing two locally optimal solutions is NP-hard for various unweighted problems, extending the understanding of local search complexity beyond the single-solution case.

Findings

Computing two local optima is NP-hard for Maximum Independent Set.

NP-hardness also applies to Minimum Dominating Set, Max SAT, and Max Cut.

Several tractable cases for multiple local optima are discussed.

Abstract

The class PLS (Polynomial Local Search) captures the complexity of finding a solution that is locally optimal and has proven to be an important concept in the theory of local search. It has been shown that local search versions of various combinatorial optimization problems, such as Maximum Independent Set and Max Cut, are complete for this class. Such computational intractability typically arises in local search problems allowing arbitrary weights; in contrast, for unweighted problems, locally optimal solutions can be found in polynomial time under standard settings. In this paper, we pursue the complexity of local search problems from a different angle: We show that computing two locally optimal solutions is NP-hard for various natural unweighted local search problems, including Maximum Independent Set, Minimum Dominating Set, Max SAT, and Max Cut. We also discuss several tractable cases for finding two (or more) local optimal solutions.