Minimum Weight Cycle (MWC) is the problem of finding a simple cycle of minimum weight in a graph $G=(V,E)$. This is a fundamental graph problem with classical sequential algorithms that run in $\tilde{O}(n^3)$ and $\tilde{O}(mn)$ time where $n=|V|$ and $m=|E|$. In recent years this problem has received significant attention in the context of fine-grained sequential complexity as well as in the design of faster sequential approximation algorithms, though not much is known in the distributed CONGEST model. We present sublinear-round approximation algorithms for computing MWC in directed graphs, and weighted graphs. Our algorithms use a variety of techniques in non-trivial ways, such as in our approximate directed unweighted MWC algorithm that efficiently computes BFS from all vertices restricted to certain implicitly computed neighborhoods in sublinear rounds, and in our weighted approximation algorithms that use unweighted MWC algorithms on scaled graphs combined with a fast and streamlined method for computing multiple source approximate SSSP. We present $\tilde{\Omega}(\sqrt{n})$ lower bounds for arbitrary constant factor approximation of MWC in directed graphs and undirected weighted graphs.

翻译：最小权重环（MWC）问题旨在图$G=(V,E)$中寻找权重最小的简单环。这是一个基础的图论问题，经典的串行算法时间复杂度为$\tilde{O}(n^3)$和$\tilde{O}(mn)$，其中$n=|V|$，$m=|E|$。近年来，该问题在细粒度串行复杂性研究以及更快的串行近似算法设计领域受到广泛关注，然而在分布式CONGEST模型中的研究尚不充分。本文提出了在有向图与加权图中计算MWC的亚线性轮次近似算法。我们的算法综合运用了多种非平凡技术：例如在近似有向无权MWC算法中，我们通过亚线性轮次高效计算所有顶点在特定隐式邻域内的BFS；在加权近似算法中，我们通过对缩放图应用无权MWC算法，并结合一种快速简化的多源近似SSSP计算方法。针对有向图与无向加权图中任意常数因子近似的MWC问题，我们给出了$\tilde{\Omega}(\sqrt{n})$的下界。