Faster approximation algorithms for computing shortest cycles on weighted graphs

Abstract

Given an n-vertex m-edge graph G with non negative edge-weights, a shortest cycle of G is one minimizing the sum of the weights on its edges. The girth of G is the weight of such a shortest cycle. We obtain several new approximation algorithms for computing the girth of weighted graphs:• For any graph G with polynomially bounded integer weights, we present a deterministic algorithm that computes, in Õ(n 5/3 + m)-time, a cycle of weight at most twice the girth of G. This matches both the approximation factor and -almost -the running time of the best known subquadratic-time approximation algorithm for the girth of unweighted graphs (Roditty and Vassilevska Williams, SODA'12). 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 + ε)-approximation for graphs with arbitrary non negative edge-weights, at the price of a slightly worse running-time in Õ(n 5/3 polylog(1/ε) + m). For that we introduce a novel polynomial-factor approximation of the girth, that makes more amenable the passing from the graphs with bounded integer edge-weights to the general case and is of independent interest.• Finally, if we insist in removing the dependency in the number m of edges, we can transform our algorithms into an Õ(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 Õ(n 5/3 )-time deterministic 4-approximation for the graphs with polynomially bounded integer weights, and an Õ(n 5/3 polylog(1/ε))-time deterministic (4 + ε)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.