  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 fine-grained sequential
complexity as well as in the design of faster sequential approximation
algorithms, though not much is known in the distributed CONGEST model.
  We present sublinear-round approximation algorithms for computing MWC in
directed graphs, and weighted graphs. 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. We present $\tilde{\Omega}(\sqrt{n})$ lower bounds for
arbitrary constant factor approximation of MWC in directed graphs and
undirected weighted graphs.
