IMPROVED GIRTH APPROXIMATION IN WEIGHTED UNDIRECTED GRAPHS*

Let G=(V,E,ℝ+) be an n-node m-edge weighted undirected graph, where ℝ+:E→(0,∞) is a real length function defined on its edges, and let g denote the girth of G, i.e., the length of a shortest cycle. We present an algorithm that, for any input, integer k≥ 1, in O(kn^1+1/k log n + m(k + log n)) expected time finds a cycle of length at most 4k/3g. This algorithm nearly matches an O(n^1+1/k log n)-time algorithm of Kadria et al. [Algorithmic trade-offs for girth approximation in undirected graphs, in Proceedings of the 2022 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), SIAM, 2022, pp. 1471–1492] which applied to unweighted graphs of girth 3. For weighted graphs, this result also improves upon the previous state-of-the-art algorithm that in O((n^1+1/k log n + m) log(nM)) time, where ℝ+:E→[1,M] is an integral length function, finds a cycle of length at most 2kg of Kadria et al. [Algorithmic trade-offs for girth approximation in undirected graphs, in Proceedings of the 2022 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), SIAM, 2022, pp. 1471–1492]. For k=1, this result improves upon the result of Roditty and Tov [ACM Trans. Algorithms, 9 (2013), pp. 15:1–15:13].