Improved girth approximation in weighted undirected graphs

TL;DR

This paper introduces an efficient algorithm for approximating the girth in weighted undirected graphs, achieving better approximation ratios and running times than previous methods, especially for weighted cases.

Contribution

The paper presents a new algorithm that approximates the girth within a factor of 4/3 in weighted graphs, improving upon previous algorithms in terms of efficiency and approximation quality.

Findings

Runs in expected time $O(kn^{1+1/k} ext{log}n + m(k+ ext{log}n))$

Provides a cycle of length at most $rac{4k}{3}g$

Improves approximation ratios for weighted graphs compared to prior work

Abstract

Let $G = (V,E,\ell)$ be a $n$-node $m$-edge weighted undirected graph, where $\ell: E \rightarrow (0,\infty)$ is a real \emph{length} function defined on its edges, and let $g$ denote the girth of $G$, i.e., the length of its shortest cycle. We present an algorithm that, for any input, integer $k \geq 1$, in $O(kn^{1+1/k}\log{n} + m(k+\log{n}))$ expected time finds a cycle of length at most $\frac{4k}{3}g$. This algorithm nearly matches a $O(n^{1+1/k}\log{n})$-time algorithm of \cite{KadriaRSWZ22} which applied to unweighted graphs of girth $3$. For weighted graphs, this result also improves upon the previous state-of-the-art algorithm that in $O((n^{1+1/k}\log n+m)\log (nM))$ time, where $\ell: E \rightarrow [1, M]$ is an integral length function, finds a cycle of length at most $2kg$~\cite{KadriaRSWZ22}. For $k=1$ this result improves upon the result of Roditty and Tov~\cite{RodittyT13}.