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 $Ω(\log n)$ rounds in the deterministic LOCAL model and $Ω(\log \log n)$ 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 $\log^{Θ(1)} n$. Our algorithms also apply to the general case of graphs of maximum degree $Δ$. For the special case of locally optimal cut, we obtain a randomized algorithm that runs in $O(Δ^{2} \log^{6} n)$ rounds, which can be derandomized at polylogarithmic cost with standard techniques. Furthermore, we show that a dependence in $Δ$ is necessary: we prove a lower bound of $Ω(\min\{Δ,\sqrt{n}\})$ rounds, even in the quantum-LOCAL model; in particular, there is no polylogarithmic-round algorithm for the general case.

翻译：本文提出了一种用于局部势能问题的快速分布式算法：这类图问题的任务是寻找局部最优解，使得任意节点无法通过改变自身标签在其局部邻域内单方面提升效用。此类问题的一个简单示例是寻找局部最优割，即对于每个节点，至少有一半关联边被割断的割。局部最优割问题的分布式轮次复杂度一直悬而未决；已知该问题在确定性LOCAL模型中需要$Ω(\log n)$轮，在随机化LOCAL模型中需要$Ω(\log \log n)$轮，但当前已知的唯一上界是$O(n)$轮的暴力解法。在常数度图中，局部最优割可能是局部可验证标记问题中最简单的例子，其当前上下界之间仍存在巨大差距。我们证明在常数度图中，所有局部势能问题（包括局部最优割）均可在$\log^{O(1)} n$轮内求解，该结论同时适用于确定性及随机化LOCAL模型。特别地，局部最优割问题的确定性轮次复杂度现可确定为$\log^{Θ(1)} n$。我们的算法同样适用于最大度为$Δ$的一般图情形。针对局部最优割这一特例，我们获得了$O(Δ^{2} \log^{6} n)$轮的随机化算法，该算法可通过标准技术以多对数代价实现去随机化。此外，我们证明对$Δ$的依赖是必要的：即使在量子LOCAL模型中，我们给出了$Ω(\min\{Δ,\sqrt{n}\})$轮的下界；特别地，一般情形不存在多对数轮算法。