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Minimum Stable Cut and Treewidth
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DOI：10.46298/dmtcs.10900.png)
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Abstract
En 中文
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 (O & centerdot; W)O(tw)nO(1), where tw is the treewidth, O the maximum degree, and W the maximum weight. On the other hand, bounding O 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 O and obtain an FPT algorithm running in time 2O(triangle 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 2o(triangle 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 + epsilon). 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 OO(tw)nO(1). We show that this is also probably essentially optimal: an algorithm running in no(pw) would contradict the ETH.
Keywords:
Treewidth
Local Max-Cut
Nash Stability
Journal
D
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0.5
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3
Citations:
485