  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 $(\Delta\cdot W)^{O(tw)}n^{O(1)}$, 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 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.
