{"paper":{"title":"Short Cycles via Low-Diameter Decompositions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.DS","authors_text":"Sushant Sachdeva, Yang P. Liu, Zejun Yu","submitted_at":"2018-10-11T17:46:48Z","abstract_excerpt":"We present improved algorithms for short cycle decomposition of a graph. Short cycle decompositions were introduced in the recent work of Chu et al, and were used to make progress on several questions in graph sparsification.\n  For all constants $\\delta \\in (0,1]$, we give an $O(mn^\\delta)$ time algorithm that, given a graph $G,$ partitions its edges into cycles of length $O(\\log n)^\\frac{1}{\\delta}$, with $O(n)$ extra edges not in any cycle. This gives the first subquadratic, in fact almost linear time, algorithm achieving polylogarithmic cycle lengths. We also give an $m \\cdot \\exp(O(\\sqrt{\\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1810.05143","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"}