Computing 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 Õ ( n 3 ) and Õ ( 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 [3, 50] as well as in the design of faster sequential approximation algorithms [13, 26, 32, 33], though not much is known in the distributed CONGEST model. We present near-optimal Ω( n ) CONGEST lower bounds on the round complexity of computing exact and (2 − ϵ)-approximate MWC in undirected weighted graphs and in directed graphs even if unweighted. We complement these lower bounds with sublinear-round algorithms for computing 2-approximation of MWC. 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.