Improved Approximation Bounds for Minimum Weight Cycle in the CONGEST Model

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 $k=n$.