Improved girth approximation in weighted undirected graphs.

Let G = (V,E,ℓ) be a n-nodes m-edges weighted undirected graph, where ℓ : E → (0, ∞) is a real length function defined on its edges. Let g be the length of the shortest cycle in G.We present an algorithm that in O(kn1+1/k log n + m(k + log n)) expected running time finds a cycle of length at most , for every integer k ≥ 1. This improves upon the previous best algorithm that in O((n1+1/k log n + m) log(nM)) time, where ℓ : E → [1, M] is an integral length function, finds a cycle of length at most 2kg [KRS+22]. For k = 1 our algorithm also improves the result of Roditty and Tov [RT13].