{"paper":{"title":"All-Pairs 2-Reachability in $\\mathcal{O}(n^{\\omega}\\log n)$ Time","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.DS","authors_text":"Daniel Graf, Giuseppe F. Italiano, Loukas Georgiadis, Nikos Parotsidis, Przemys{\\l}aw Uzna\\'nski","submitted_at":"2016-12-23T20:05:32Z","abstract_excerpt":"In the $2$-reachability problem we are given a directed graph $G$ and we wish to determine if there are two (edge or vertex) disjoint paths from $u$ to $v$, for a given pair of vertices $u$ and $v$. In this paper, we present an algorithm that computes $2$-reachability information for all pairs of vertices in $\\mathcal{O}(n^{\\omega}\\log n)$ time, where $n$ is the number of vertices and $\\omega$ is the matrix multiplication exponent. Hence, we show that the running time of all-pairs $2$-reachability is only within a $\\log$ factor of transitive closure.\n  Moreover, our algorithm produces a witnes"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1612.08075","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"}