Faster Approximation Algorithms for Computing Shortest Cycles on Weighted Graphs

Given an $n$-vertex $m$-edge graph $G$ with nonnegative edge-weights, a shortest cycle is one minimizing the sum of the weights on its edges. The girth of $G$ is the weight of 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 $\tilde{\cal O}(n^{5/3}+m)$-time, a cycle of weight at most twice the girth of $G$. This matches the approximation factor of the best known subquadratic-time approximation algorithm for the girth of unweighted graphs. Then, we turn our algorithm into a deterministic $(2+\varepsilon)$-approximation for graphs with arbitrary nonnegative edge-weights, at the price of a slightly worse running time in $\tilde{\cal O}(n^{5/3}\text{polylog}{(1/\varepsilon)}+m)$. 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 $\tilde{\cal O}(n^{5/3})$-time randomized 4-approximation for graphs with nonnegative edge-weights. This can be derandomized, thereby leading to an $\tilde{\cal O}(n^{5/3})$-time deterministic 4-approximation for graphs with polynomially bounded integer weights, and an $\tilde{\cal O}(n^{5/3}\text{polylog}{(1/\varepsilon)})$-time deterministic $(4+\varepsilon)$-approximation for graphs with nonnegative edge-weights. To the best of our knowledge, these are the first known subquadratic-time approximation algorithms for computing the girth of weighted graphs.