The Complexity of Distributed Minimum Weight Cycle Approximation

We investigate the \emph{minimum weight cycle (MWC)} problem in the $\mathsf{CONGEST}$ model of distributed computing. For undirected weighted graphs, we design a randomized algorithm that achieves a $(k+1)$-approximation, for any \emph{real} number $k \ge 1$. The round complexity of algorithm is $$\tilde{O}\!\Big( n^{\frac{k+1}{2k+1}} + n^{\frac{1}{k}} + D\, n^{\frac{1}{2(2k+1)}} + D^{\frac{2}{5}} n^{\frac{2}{5}+\frac{1}{2(2k+1)}} \Big).$$ where $n$ denotes the number of nodes and $D$ is the unweighted diameter of the graph. This result yields a smooth trade-off between approximation ratio and round complexity. In particular, when $k \geq 2$ and $D = \tilde{O}(n^{1/4})$, the bound simplifies to $$\tilde{O}\!\left( n^{\frac{k+1}{2k+1}} \right)$$ On the lower bound side, assuming the Erd\H{o}s girth conjecture, we prove that for every \emph{integer} $k \ge 1$, any randomized $(k+1-\epsilon)$-approximation algorithm for MWC requires $$\tilde{\Omega}\!\left( n^{\frac{k+1}{2k+1}} \right)$$ rounds. This lower bound holds for both directed unweighted and undirected weighted graphs, and applies even to graphs with small diameter $D = \Theta(\log n)$. Taken together, our upper and lower bounds \emph{match up to polylogarithmic factors} for graphs of sufficiently small diameter $D = \tilde{O}(n^{1/4})$ (when $k \geq 2$), yielding a nearly tight bound on the distributed complexity of the problem. Our results improve upon the previous state of the art: Manoharan and Ramachandran (PODC~2024) demonstrated a $(2+\epsilon)$-approximation algorithm for undirected weighted graphs with round complexity $\tilde{O}(n^{2/3}+D)$, and proved that for any arbitrarily large number $\alpha$, any $\alpha$-approximation algorithm for directed unweighted or undirected weighted graphs requires $\Omega(\sqrt{n}/\log n)$ rounds.