Faster approximation algorithms for computing shortest cycles on weighted graphs

TL;DR

This paper introduces new faster approximation algorithms for computing the girth of weighted graphs, achieving subquadratic runtime with guarantees close to the optimal girth, a significant advancement over previous methods.

Contribution

The paper presents the first subquadratic-time approximation algorithms for girth in weighted graphs, combining novel insights with Hitting Set techniques for improved efficiency.

Findings

Deterministic 2-approximation in O(n^{5/3}+m) time

A O(n^{5/3}\u00b1 log^{O(1)}(1/psilon))-time (4+psilon)-approximation

Randomized O(n^{5/3})-time 4-approximation, derandomized for polynomial weights

Abstract

Given an $n$-vertex $m$-edge graph $G$ with non negative edge-weights, the girth of $G$ is the weight of a shortest cycle in $G$. For any graph $G$ with polynomially bounded integer weights, we present a deterministic algorithm that computes, in $\tilde{\cal O}(n^{5/3}+m)$-time, a cycle of weight at most twice the girth of $G$. Our approach combines some new insights on the previous approximation algorithms for this problem (Lingas and Lundell, IPL'09; Roditty and Tov, TALG'13) with Hitting Set based methods that are used for approximate distance oracles and date back from (Thorup and Zwick, JACM'05). Then, we turn our algorithm into a deterministic $(2+\varepsilon)$-approximation for graphs with arbitrary non negative edge-weights, at the price of a slightly worse running-time in $\tilde{\cal O}(n^{5/3}\log^{{\cal O}(1)}{(1/\varepsilon)}+m)$. Finally, if we insist in removing the dependency in the number $m$ of edges, we can transform our algorithms into an $\tilde{\cal O}(n^{5/3})$-time randomized $4$-approximation for the graphs with non negative edge-weights -- assuming the adjacency lists are sorted. Combined with the aforementioned Hitting Set based methods, this algorithm can be derandomized, thereby yielding an $\tilde{\cal O}(n^{5/3})$-time deterministic $4$-approximation for the graphs with polynomially bounded integer weights, and an $\tilde{\cal O}(n^{5/3}\log^{{\cal O}(1)}{(1/\varepsilon)})$-time deterministic $(4+\varepsilon)$-approximation for the graphs with non negative edge-weights. To the best of our knowledge, these are the first known subquadratic-time approximation algorithms for computing the girth of weighted graphs.