{"paper":{"title":"Approximating Cycles in Directed Graphs: Fast Algorithms for Girth and Roundtrip Spanners","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.DS","authors_text":"Aaron Sidford, Jakub Pachocki, Liam Roditty, Roei Tov, Virginia Vassilevska Williams","submitted_at":"2016-11-02T18:40:54Z","abstract_excerpt":"The girth of a graph, i.e. the length of its shortest cycle, is a fundamental graph parameter. Unfortunately all known algorithms for computing, even approximately, the girth and girth-related structures in directed weighted $m$-edge and $n$-node graphs require $\\Omega(\\min\\{n^{\\omega}, mn\\})$ time (for $2\\leq\\omega<2.373$). In this paper, we drastically improve these runtimes as follows:\n  * Multiplicative Approximations in Nearly Linear Time: We give an algorithm that in $\\widetilde{O}(m)$ time computes an $\\widetilde{O}(1)$-multiplicative approximation of the girth as well as an $\\widetilde"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1611.00721","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}