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 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 $k=n$.