{"paper":{"title":"On Weighted Multicommodity Flows in Directed Networks","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"Alexander V. Karzanov, Maxim A. Babenko","submitted_at":"2012-12-02T16:53:04Z","abstract_excerpt":"Let $G = (VG, AG)$ be a directed graph with a set $S \\subseteq VG$ of terminals and nonnegative integer arc capacities $c$. A feasible multiflow is a nonnegative real function $F(P)$ of \"flows\" on paths $P$ connecting distinct terminals such that the sum of flows through each arc $a$ does not exceed $c(a)$. Given $\\mu \\colon S \\times S \\to \\R_+$, the \\emph{$\\mu$-value} of $F$ is $\\sum_P F(P) \\mu(s_P, t_P)$, where $s_P$ and $t_P$ are the start and end vertices of a path $P$, respectively.\n  Using a sophisticated topological approach, Hirai and Koichi showed that the maximum $\\mu$-value multiflo"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1212.0224","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":""},"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"}