Minimum Stable Cut and Treewidth

Abstract

A stable or locally-optimal cut of a graph is a cut whose weight cannot be increased by changing the side of a single vertex. Equivalently, a cut is stable if all vertices have the (weighted) majority of their neighbors on the other side. Finding a stable cut is a prototypical PLS-complete problem that has been studied in the context of local search and of algorithmic game theory.In this paper we study Min Stable Cut, the problem of finding a stable cut of minimum weight, which is closely related to the Price of Anarchy of the Max Cut game. Since this problem is NP-hard, we study its complexity on graphs of low treewidth, low degree, or both. We begin by showing that the problem remains weakly NP-hard on severely restricted trees, so bounding treewidth alone cannot make it tractable. We match this hardness with a pseudo-polynomial DP algorithm solving the problem in time (∆ • W ) O(tw) n O(1) , where tw is the treewidth, ∆ the maximum degree, and W the maximum weight. On the other hand, bounding ∆ is also not enough, as the problem is NP-hard for unweighted graphs of bounded degree. We therefore parameterize Min Stable Cut by both tw and ∆ and obtain an FPT algorithm running in time 2 O(∆tw) (n + log W ) O(1) . Our main result for the weighted problem is to provide a reduction showing that both aforementioned algorithms are essentially optimal, even if we replace treewidth by pathwidth: if there exists an algorithm running in (nW ) o(pw) or 2 o(∆pw) (n + log W ) O(1) , then the ETH is false. Complementing this, we show that we can, however, obtain an FPT approximation scheme parameterized by treewidth, if we consider almost-stable solutions, that is, solutions where no single vertex can unilaterally increase the weight of its incident cut edges by more than a factor of (1 + ε).Motivated by these mostly negative results, we consider Unweighted Min Stable Cut. Here our results already imply a much faster exact algorithm running in time ∆ O(tw) n O(1) . We show that this is also probably essentially optimal: an algorithm running in n o(pw) would contradict the ETH. ACM Subject ClassificationMathematics of computing→Graph algorithms; Theory of Computation → Design and Analysis of Algorithms → Parameterized Complexity and Exact Algorithms