Minimum Stable Cut and Treewidth

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 $(\Delta\cdot W)^{O(tw)}n^{O(1)}$, where $tw$ is the treewidth, $\Delta$ the maximum degree, and $W$ the maximum weight. On the other hand, bounding $\Delta$ is also not enough, as the problem is NP-hard for unweighted graphs of bounded degree. We therefore parameterize Min Stable Cut by both $tw+\Delta$ and obtain an FPT algorithm running in time $2^{O(\Delta tw)}(n+\log W)^{O(1)}$. 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 $2^{o(\Delta pw)}(n+\log W)^{O(1)}$, 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 $\Delta^{O(tw)}n^{O(1)}$. We show that this is also probably essentially optimal: an algorithm running in $n^{o(pw)}$ would contradict the ETH.