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 $(Δ\cdot W)^{O(tw)}n^{O(1)}$, 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 Minimum Stable Cut by both $tw$ and $Δ$ and obtain an FPT algorithm running in time $2^{O(Δtw)}(n+\log W)^{O(1)}$. 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 $2^{o(Δpw)}(n+\log W)^{O(1)}$, 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 $Δ^{O(tw)}n^{O(1)}$. We show that this is also probably essentially optimal: an algorithm running in $n^{o(pw)}$ would contradict the ETH.

翻译：稳定割或局部最优割是指无法通过改变单个顶点所在集合来增加其权重的图割。本文研究最小稳定割问题，即寻找权重最小的稳定割。由于该问题属于NP难问题，我们研究了其在低树宽、低度或两者兼具图上的复杂度。首先我们证明，即使在高度受限的树上该问题仍为弱NP难，因此仅限制树宽无法使其易于求解。我们通过一个伪多项式动态规划算法与之匹配，该算法可在$(Δ\cdot W)^{O(tw)}n^{O(1)}$时间内求解，其中$tw$为树宽，$Δ$为最大度数，$W$为最大权重。另一方面，仅限制$Δ$也不够，因为该问题在有界度无权重图上仍为NP难。因此我们同时以$tw$和$Δ$为参数化标准，提出一个运行时间为$2^{O(Δtw)}(n+\log W)^{O(1)}$的固定参数可解（FPT）算法。对于带权问题，我们的主要成果是通过归约证明：即使将树宽替换为路径宽，上述两种算法本质上仍是最优的——若存在运行时间为$(nW)^{o(pw)}$或$2^{o(Δpw)}(n+\log W)^{O(1)}$的算法，则指数时间假设（ETH）不成立。作为补充，我们证明若考虑近似稳定解（即单个顶点最多能将与其关联割边权重提升$(1+\varepsilon)$倍），则可得到基于树宽参数的FPT近似方案。受这些负面结果的驱动，我们进一步研究无权重最小稳定割问题。此时我们已有结果可直接给出一个运行时间为$Δ^{O(tw)}n^{O(1)}$的快速精确算法，且该算法很可能本质上亦为最优：任何$n^{o(pw)}$时间的算法都将与ETH矛盾。