We present a complete classification of the distributed computational complexity of local optimization problems in directed cycles for both the deterministic and the randomized LOCAL model. We show that for any local optimization problem $Π$ (that can be of the form min-sum, max-sum, min-max, or max-min, for any local cost or utility function over some finite alphabet), and for any constant approximation ratio $α$, the task of finding an $α$-approximation of $Π$ in directed cycles has one of the following complexities: 1. $O(1)$ rounds in deterministic LOCAL, $O(1)$ rounds in randomized LOCAL, 2. $Θ(\log^* n)$ rounds in deterministic LOCAL, $O(1)$ rounds in randomized LOCAL, 3. $Θ(\log^* n)$ rounds in deterministic LOCAL, $Θ(\log^* n)$ rounds in randomized LOCAL, 4. $Θ(n)$ rounds in deterministic LOCAL, $Θ(n)$ rounds in randomized LOCAL. Moreover, for any given $Π$ and $α$, we can determine the complexity class automatically, with an efficient (centralized, sequential) meta-algorithm, and we can also efficiently synthesize an asymptotically optimal distributed algorithm. Before this work, similar results were only known for local search problems (e.g., locally checkable labeling problems). The family of local optimization problems is a strict generalization of local search problems, and it contains numerous commonly studied distributed tasks, such as the problems of finding approximations of the maximum independent set, minimum vertex cover, minimum dominating set, and minimum vertex coloring.

翻译：我们针对确定性及随机化LOCAL模型，对有向环中局部优化问题的分布式计算复杂度进行了完整分类。我们证明，对于任意局部优化问题$Π$（其形式可为最小和、最大和、最小最大值或最大最小值问题，涉及定义在有限字母表上的任意局部成本或效用函数），以及任意常数近似比$α$，在有向环中寻找$Π$的$α$近似解的任务具有以下复杂度之一：1. 确定性LOCAL模型中$O(1)$轮，随机化LOCAL模型中$O(1)$轮；2. 确定性LOCAL模型中$Θ(\log^* n)$轮，随机化LOCAL模型中$O(1)$轮；3. 确定性LOCAL模型中$Θ(\log^* n)$轮，随机化LOCAL模型中$Θ(\log^* n)$轮；4. 确定性LOCAL模型中$Θ(n)$轮，随机化LOCAL模型中$Θ(n)$轮。此外，对于任意给定的$Π$和$α$，我们可以通过高效的（集中式、顺序执行的）元算法自动判定其所属的复杂度类别，并能高效合成渐近最优的分布式算法。在本研究之前，类似结论仅针对局部搜索问题（例如局部可检查标记问题）已知。局部优化问题族是局部搜索问题的严格推广，包含大量常见分布式任务，例如寻找最大独立集、最小顶点覆盖、最小支配集及最小顶点着色等问题的近似解。