In this work, we present a fast distributed algorithm for local potential problems: these are graph problems where the task is to find a locally optimal solution where no node can unilaterally improve the utility in its local neighborhood by changing its own label. A simple example of such a problem is the task of finding a locally optimal cut, i.e., a cut where for each node at least half of its incident edges are cut edges.

The distributed round complexity of the locally optimal cut problem has been wide open; the problem is known to require $\Omega(\log n)$ rounds in the deterministic LOCAL model and $\Omega(\log \log n)$ rounds in the randomized LOCAL model, but the only known upper bound is the trivial brute-force solution of $O(n)$ rounds. Locally optimal cut in constant-degree graphs is perhaps the simplest example of a locally checkable labeling problem for which there is still such a large gap between current upper and lower bounds.

We show that in constant-degree graphs, all local potential problems, including locally optimal cut, can be solved in $\log^{O(1)} n$ rounds, both in the deterministic and randomized LOCAL models. In particular, the deterministic round complexity of the locally optimal cut problem is now settled to $\log^{\Theta(1)} n$.

Our algorithms also apply to the general case of graphs of maximum degree $\Delta$. For the special case of locally optimal cut, we obtain a randomized algorithm that runs in $O(\Delta^2 \log^6 n)$ rounds, which can be derandomized at polylogarithmic cost with standard techniques. Furthermore, we show that a dependence in $\Delta$ is necessary: we prove a lower bound of $\Omega(\min\{\Delta,\sqrt{n}\})$ rounds, even in the quantum-LOCAL model; in particular, there is no polylogarithmic-round algorithm for the general case.