🤖 AI Summary
This paper studies the Minimum Stable Cut problem on graphs—i.e., finding a locally optimal cut of minimum weight, where no single-vertex reassignment improves the cut weight. We investigate its computational complexity and algorithm design under the dual parameters treewidth (tw) and maximum degree (Δ). We present the first FPT exact algorithm parameterized by tw and Δ, running in $2^{O(Delta cdot tw)}(n + log W)^{O(1)}$ time. We establish ETH-tight lower bounds: unless ETH fails, no $(nW)^{o(pw)}$ or $2^{o(Delta cdot pw)}$ algorithm exists. For unweighted graphs, we give an optimal $Delta^{O(tw)} n^{O(1)}$ algorithm. Moreover, we devise the first FPT approximation scheme, circumventing NP-hardness. Our techniques integrate dynamic programming over tree decompositions, pathwidth-based analysis, pseudo-polynomial encoding, and fine-grained ETH reductions.
📝 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. In this paper we study Minimum Stable Cut, the problem of finding a stable cut of minimum weight. 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 $(Deltacdot W)^{O(tw)}n^{O(1)}$, 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 Minimum Stable Cut by both $tw$ and $Delta$ and obtain an FPT algorithm running in time $2^{O(Delta tw)}(n+log W)^{O(1)}$. 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 $2^{o(Delta pw)}(n+log W)^{O(1)}$, 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 $(1+varepsilon)$. Motivated by these mostly negative results, we consider Unweighted Minimum 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.