Parameterized Local Search for Vertex Cover: When Only the Search Radius Is Crucial

Abstract

A k-swap W for a vertex cover S of a graph G is a vertex set of size at most k such that S\u27 = (S ? W) ? (W ? S), the symmetric difference of S and W, is a vertex cover of G. If |S\u27| < |S|, then W is improving. In LS-Vertex Cover, one is given a vertex cover S of a graph G and wants to know if there is an improving k-swap for S in G. In applications of LS-Vertex Cover, k is a very small parameter that can be set by a user to determine the trade-off between running time and solution quality. Consequently, k can be considered to be a constant. Motivated by this and the fact that LS-Vertex Cover is W[1]-hard with respect to k, we aim for algorithms with running time ?^f(k) ? n^?(1) where ? is a structural graph parameter upper-bounded by n. We say that such a running time grows mildly with respect to ? and strongly with respect to k. We obtain algorithms with such a running time for ? being the h-index of G, the treewidth of G, or the modular-width of G. In addition, we consider a novel parameter, the maximum degree over all quotient graphs in a modular decomposition of G. Moreover, we adapt these algorithms to the more general problem where each vertex is assigned a weight and where we want to find a d-improving k-swap, that is, a k-swap which decreases the weight of the vertex cover by at least d