Faster Approximation Algorithms for computing shortest cycles on weighted graphs

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 O~(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. - Then, we turn our algorithm into a deterministic (2+epsilon)-approximation for graphs with arbitrary non-negative edge-weights, at the price of a slightly worse running-time in O~(n^{5/3}polylog(1/epsilon)+m). For that, we introduce a generic method in order to obtain a polynomial-factor approximation of the girth in subquadratic time, that may be of independent interest. - Finally, if we assume that the adjacency lists are sorted then we can get rid off the dependency in the number m of edges. Namely, we can transform our algorithms into an O~(n^{5/3})-time randomized 4-approximation for graphs with non-negative edge-weights. This can be derandomized, thereby leading to an O~(n^{5/3})-time deterministic 4-approximation for graphs with polynomially bounded integer weights, and an O~(n^{5/3}polylog(1/epsilon))-time deterministic (4+epsilon)-approximation for 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.