Don't Be Strict in Local Search!

TL;DR

This paper explores a relaxed version of local search called permissive local search, demonstrating that it can efficiently solve certain instances of the Vertex Cover problem where strict local search remains computationally hard, within the framework of parameterized complexity.

Contribution

It introduces and analyzes permissive local search for Vertex Cover, showing it can be tractable in cases where strict local search is hard, advancing understanding of local search relaxations.

Findings

Permissive local search is tractable for certain hard instances of Vertex Cover.

Strict local search remains computationally hard for the same instances.

Relaxing local search constraints can expand the domain of tractable problems.

Abstract

Local Search is one of the fundamental approaches to combinatorial optimization and it is used throughout AI. Several local search algorithms are based on searching the k-exchange neighborhood. This is the set of solutions that can be obtained from the current solution by exchanging at most k elements. As a rule of thumb, the larger k is, the better are the chances of finding an improved solution. However, for inputs of size n, a na\"ive brute-force search of the k-exchange neighborhood requires n to the power of O(k) time, which is not practical even for very small values of k. Fellows et al. (IJCAI 2009) studied whether this brute-force search is avoidable and gave positive and negative answers for several combinatorial problems. They used the notion of local search in a strict sense. That is, an improved solution needs to be found in the k-exchange neighborhood even if a global optimum can be found efficiently. In this paper we consider a natural relaxation of local search, called permissive local search (Marx and Schlotter, IWPEC 2009) and investigate whether it enhances the domain of tractable inputs. We exemplify this approach on a fundamental combinatorial problem, Vertex Cover. More precisely, we show that for a class of inputs, finding an optimum is hard, strict local search is hard, but permissive local search is tractable. We carry out this investigation in the framework of parameterized complexity.