The Complexity of Distributed Minimum Weight Cycle Approximation

2026-03-26

AI Summary

This work studies the Minimum Weight Cycle (MWC) problem in undirected weighted graphs within the distributed CONGEST model. It proposes a randomized algorithm parameterized by an integer $k \geq 1$, achieving a smooth trade-off between approximation ratio and round complexity: for any $k$, it yields a $(k+1)$-approximation in $\widetilde{O}(n^{(k+1)/(2k+1)} + n^{1/k} + D)$ rounds, where $D$ is the graph diameter. By combining a reduction based on Erdős’s girth conjecture with graph decomposition techniques, the paper establishes nearly tight upper and lower bounds—differing by at most logarithmic factors—and characterizes the intrinsic distributed complexity of MWC, particularly for small-diameter graphs (e.g., $D = \widetilde{O}(n^{1/4})$ with $k \geq 2$), thereby overcoming prior limitations restricted to fixed approximation ratios.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Planning, Routing, and Scheduling: Scheduling under Uncertainty
• Search and Optimization: Distributed Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Content-based information diffusion
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

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 Erdős girth conjecture, we prove that for every \emph{integer} $k \ge 1$, any randomized $(k+1-ε)$-approximation algorithm for MWC requires \[ \tildeΩ\!\left( n^{\frac{k+1}{2k+1}} \right) \] rounds. This lower bound holds for both directed unweighted and undirected weighted graphs, and applies even to graphs with small diameter $D = Θ(\log n)$. 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 $(2+ε)$-approximation algorithm for undirected weighted graphs with round complexity $\tilde{O}(n^{2/3}+D)$, and proved that for any arbitrarily large number $α$, any $α$-approximation algorithm for directed unweighted or undirected weighted graphs requires $Ω(\sqrt{n}/\log n)$ rounds.

Problem

• minimum weight cycle
• distributed computing
• CONGEST model
• approximation algorithm
• round complexity

Innovation

• minimum weight cycle
• distributed approximation
• CONGEST model
• round complexity
• tight bounds