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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1212.0224","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-12-02T16:53:04Z","cross_cats_sorted":["cs.DM"],"title_canon_sha256":"7660ec5286995cbfa5477d57f65cecc72eda6d5025df51ecb6823aeb02f535c2","abstract_canon_sha256":"4b82767ab5484de0ed82874ce0a2568132690d0c73385559ffeeca18c631bdaa"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:39:26.836414Z","signature_b64":"ZHvS0x2iiLid+gJR7vjRumAOsaUdwlyR3diahGoU18R/rr1qhkreDksOBdyb4GdRJqIJubTU6OlzhlgCf7UYBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"33028639ba4a65a0e7a2904d072ef5b17960778045fa610724211864a94d0dbd","last_reissued_at":"2026-05-18T03:39:26.835616Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:39:26.835616Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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. 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