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 \emph{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：本文针对确定性与随机性LOCAL模型，完整地分类了有向环中局部优化问题的分布式计算复杂度。我们证明，对于任意局部优化问题$Π$（其形式可以是基于有限字母表上局部成本或效用函数的min-sum、max-sum、min-max或max-min问题），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中$O(1)$轮，随机性LOCAL中$O(1)$轮；2. 确定性LOCAL中$Θ(\log^* n)$轮，随机性LOCAL中$O(1)$轮；3. 确定性LOCAL中$Θ(\log^* n)$轮，随机性LOCAL中$Θ(\log^* n)$轮；4. 确定性LOCAL中$Θ(n)$轮，随机性LOCAL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、串行）元算法自动判定其所属复杂度类，并能高效合成渐近最优的分布式算法。在本研究之前，类似结果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，包含众多常见分布式任务，例如寻找最大独立集、最小顶点覆盖、最小支配集和最小顶点着色等问题的近似解。