Girth Approximations in the CONGEST Model

TL;DR

This paper presents new algorithms and bounds for girth approximation in the CONGEST model, improving efficiency and narrowing complexity gaps for various network types.

Contribution

It introduces a unified family of algorithms with tradeoffs for undirected networks, improves bounds for directed networks, and establishes tighter lower bounds.

Findings

Algorithms for girth approximation with specific round complexities

Improved bounds for directed and weighted networks

New lower bounds assuming Erdős-Simonovits' cycle conjecture

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 Erd\H{o}s-Simonovits' even cycle conjecture (and unconditionally for $k\in\{3,4,6\}$), any $(k-\varepsilon)$-approximation for the girth requires $\tilde{\Omega}(n^{k/(2k-1)})$ rounds, even when $D = O(\log n)$.