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 hardness through fine grained sequential complexity as well as in design of faster sequential approximation algorithms. For computing minimum weight cycle in the distributed CONGEST model, near-linear in $n$ lower and upper bounds on round complexity are known for directed graphs (weighted and unweighted), and for undirected weighted graphs; these lower bounds also apply to any $(2-\epsilon)$-approximation algorithm. This paper focuses on round complexity bounds for approximating MWC in the CONGEST model: For coarse approximations we show that for any constant $\alpha >1$, computing an $\alpha$-approximation of MWC requires $\Omega (\frac{\sqrt n}{\log n})$ rounds on weighted undirected graphs and on directed graphs, even if unweighted. We complement these lower bounds with a sublinear $\tilde{O}(n^{2/3}+D)$-round algorithm to compute a $(2+\epsilon)$-approximation of undirected weighted MWC. We also give a $\tilde{O}(n^{4/5}+D)$-round algorithm to compute 2-approximate directed unweighted MWC and $(2+\epsilon)$-approximate directed weighted MWC. To obtain the sublinear round bounds, we design an efficient algorithm for computing $(1+\epsilon)$-approximate shortest paths from $k$ sources in directed and weighted graphs, which is of independent interest. We present an algorithm that runs in $\tilde{O}(\sqrt{nk} + D)$ rounds if $k \ge n^{1/3}$ and $\tilde{O}(\sqrt{nk} + k^{2/5}n^{2/5+o(1)}D^{2/5} + D)$ rounds if $k<n^{1/3}$, which smoothly interpolates between the best known upper bounds for SSSP when $k=1$ and APSP when $k=n$.

翻译：最小权环（MWC）是寻找图$G=(V,E)$中最小权简单环的问题。这是一个基础图论问题，经典顺序算法可在$\tilde{O}(n^3)$和$\tilde{O}(mn)$时间内求解，其中$n=|V|$，$m=|E|$。近年来，该问题在细粒度顺序复杂度下的困难性以及更快速顺序近似算法的设计方面受到广泛关注。在分布式CONGEST模型中计算最小权环时，有向图（加权与未加权）和无向加权图的轮数复杂度已知近线性于$n$的下界与上界；这些下界同样适用于任何$(2-\epsilon)$近似算法。本文聚焦于CONGEST模型中近似MWC的轮数复杂度：对于粗略近似，我们证明对任意常数$\alpha >1$，即使在无向加权图和有向图（包括未加权情况）上计算$\alpha$近似MWC也需要$\Omega (\frac{\sqrt n}{\log n})$轮。针对这些下界，我们设计了亚线性$\tilde{O}(n^{2/3}+D)$轮算法，用于计算无向加权MWC的$(2+\epsilon)$近似。此外，我们给出$\tilde{O}(n^{4/5}+D)$轮算法，分别计算2近似有向未加权MWC和$(2+\epsilon)$近似有向加权MWC。为获得亚线性轮数界，我们设计了一种高效算法，用于在有向加权图中从$k$个源点计算$(1+\epsilon)$近似最短路径，该问题本身具有独立研究价值。我们提出的算法在$k \ge n^{1/3}$时运行时间为$\tilde{O}(\sqrt{nk} + D)$轮，在$k<n^{1/3}$时为$\tilde{O}(\sqrt{nk} + k^{2/5}n^{2/5+o(1)}D^{2/5} + D)$轮，该结果平滑地插值了已知最优上界：当$k=1$时对应SSSP，当$k=n$时对应APSP。