Girth Approximations in the CONGEST Model

2026-03-29

AI Summary

This work systematically investigates girth approximation in undirected, directed, and weighted graphs under the CONGEST model, focusing on the trade-off between round complexity and approximation ratio. It introduces a unified algorithmic framework that yields, for the first time, a flexible family of algorithms for undirected graphs balancing approximation quality and communication rounds. The study improves the approximation ratio for directed graphs to 2 or (2+ε), and establishes a tighter lower bound of Ω̃(n^{k/(2k−1)}) rounds based on the Erdős–Simonovits even-cycle conjecture. Concrete results include an f-approximation in Õ(n^{1/f}+D) rounds for unweighted undirected graphs, a (2k−1+o(1))-approximation in Õ(n^{(k+1)/(2k+1)}+D) rounds for weighted undirected graphs, and an Õ(n^{2/3}+D)-round algorithm for directed graphs, significantly narrowing the complexity gaps across multiple graph classes.

Abstract

This paper advances the state of the art in girth approximation within the CONGEST model. Manoharan and Ramachandran [PODC '24] provided the first significant improvement in girth approximation in over a decade. We build on this momentum and make progress on all fronts: we provide a unified family of algorithms yielding girth approximation-round tradeoffs for undirected networks; we obtain improved bounds for directed networks; and we establish better lower bounds for directed and undirected weighted networks. Together, these results substantially narrow the remaining complexity gaps across all settings. Specifically, for networks with $n$ nodes and hop-diameter $D$, we show that one can compute, with high probability: $(1)$ An $f$-approximation for unweighted undirected girth in $\tilde{O}(n^{1/f}+D)$ rounds, for every constant integer $f>2$, $(2)$ A $(2k-1+o(1))$-approximation for weighted undirected girth in $\tilde{O}(n^{(k+1)/(2k+1)}+D)$ rounds, for every constant integer $k>1$, and $(3)$ A $2$-approximation for directed unweighted girth, and a $(2+\varepsilon)$-approximation for directed weighted girth, both in $\tilde{O}(n^{2/3}+D)$ rounds. We also prove new lower bounds for directed networks and for undirected weighted networks: for every integer $k > 2$ and $\varepsilon>0$, assuming the Erdős-Simonovits' even cycle conjecture (and unconditionally for $k\in\{3,4,6\}$), any $(k-\varepsilon)$-approximation for the girth requires $\tildeΩ(n^{k/(2k-1)})$ rounds, even when $D = O(\log n)$.

Problem

• girth
• CONGEST model
• distributed algorithms
• graph approximation
• round complexity

Innovation

• girth approximation
• CONGEST model
• distributed algorithms
• round complexity
• lower bounds