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 sublinear $\tilde{O}(n^{2/3}+D)$-round algorithms for approximating MWC close to a factor of 2 in these classes of graphs. A key ingredient of our approximation algorithms is an efficient algorithm for computing $(1+\epsilon)$-approximate shortest paths from $k$ sources in directed and weighted graphs, which may be of independent interest for other CONGEST problems. 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 $k<n^{1/3}$, and this round complexity smoothly interpolates between the best known upper bounds for approximate (or exact) SSSP when $k=1$ and APSP when $k=n$.

翻译：最小权重环（MWC）问题是指在图$G=(V,E)$中寻找权重最小的简单环。这是一个基础的图论问题，其经典顺序算法的时间复杂度分别为$\tilde{O}(n^3)$和$\tilde{O}(mn)$，其中$n=|V|$，$m=|E|$。近年来，该问题在细粒度顺序复杂性的难解性分析以及更快顺序近似算法的设计方面受到广泛关注。针对分布式CONGEST模型中最小权重环的计算，有向图（加权与无权）以及无向加权图的轮次复杂度已知达到近线性$n$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的轮次复杂度界：对于粗粒度近似，我们证明对任意常数$\alpha >1$，在加权无向图和有向图（即使是无权图）上，计算MWC的$\alpha$-近似需要$\Omega (\frac{\sqrt n}{\log n})$轮。针对这些图类，我们提出亚线性的$\tilde{O}(n^{2/3}+D)$轮次算法，可实现接近2倍因子的MWC近似。我们近似算法的关键要素是，在有向加权图中从$k$个源点计算$(1+\epsilon)$-近似最短路径的高效算法——该算法可能对其他CONGEST问题具有独立价值。我们提出的算法在$k \ge n^{1/3}$时运行于$\tilde{O}(\sqrt{nk} + D)$轮，在$k<n^{1/3}$时运行于$\tilde{O}(\sqrt{nk} + k^{2/5}n^{2/5+o(1)}D^{2/5} + D)$轮，且该轮次复杂度在$k=1$时（近似SSSP）与$k=n$时（APSP）的已知最优上界之间平滑插值。