Minimum Stable Cut and Treewidth
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A stable 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. 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 show that the problem is weakly NP-hard on severely restricted trees, so bounding treewidth alone cannot make it tractable. We match this with a pseudo-polynomial DP algorithm running in time (Δ· W)^O(tw)n^O(1), where tw is the treewidth, Δ the maximum degree, and W the maximum weight. On the other hand, bounding Δ 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+Δ and obtain an FPT algorithm running in time 2^O(Δ tw)(n+log W)^O(1). Our main result 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(Δ pw)(n+log W)^O(1), then the ETH is false. Complementing this, we show that we can obtain an FPT approximation scheme parameterized by treewidth, if we consider almost-stable solutions. 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 Δ^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.
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