{"paper":{"title":"Undirected 3-Fault Replacement Path in Nearly Cubic Time","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.DS","authors_text":"Benyu Wang, Ran Duan, Shucheng Chi, Tianle Xie","submitted_at":"2024-11-27T13:07:04Z","abstract_excerpt":"Given a graph $G=(V,E)$ and two vertices $s,t\\in V$, the $f$-fault replacement path ($f$FRP) problem computes for every set of edges $F$ where $|F|\\leq f$, the distance from $s$ to $t$ when edges in $F$ fail. A recent result shows that 2FRP in directed graphs can be solved in $\\tilde{O}(n^3)$ time [arXiv:2209.07016]. In this paper, we show a 3FRP algorithm in deterministic $\\tilde{O}(n^3)$ time for undirected weighted graphs, which almost matches the size of the output. This implies that $f$FRP in undirected graphs can be solved in almost optimal $\\tilde{O}(n^f)$ time for all $f\\geq 3$. To con"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2411.18312","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/2411.18312/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"}