{"paper":{"title":"Multi-Source Reachability in Near-Optimal Time","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"cs.DS","authors_text":"Merav Parter, Shimon Kogan","submitted_at":"2026-06-24T09:21:25Z","abstract_excerpt":"The multi-source reachability problem asks to compute the reachable sets from a given subset of source vertices. For $n$-vertex digraphs $G=(V,E)$ and a subset of sources $S \\subseteq V$ with $|S|=n^{\\sigma}$ for some $\\sigma \\in [0,1]$, we present a near-optimal deterministic algorithm that solves this problem in $\\tilde{O}(n^{\\omega(\\sigma)})$ time, where $\\omega(\\sigma)$ is the rectangular matrix multiplication exponent for multiplying an $n^{\\sigma}\\times n$ matrix by an $n \\times n$ matrix. For dense graphs, this yields reachability from up to $n^{0.32}$ sources in near-linear time, break"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.25612","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2606.25612/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}