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The edges $E(D)$ of $D$ are given independent edge costs $C(e),e\\in E(D)$, such that (i) $C$ has a density $f$ that satisfies $f(x)=a+bx+O(x^2)$, for constants $a>0,b$ as $x\\to 0$ and such that in general either (ii) $\\Pr(C\\geq x)\\leq \\a e^{-\\b x}$ for constants $\\a,\\b>0$, or $f(x)=0$ for $x>\\n$ for some constant $\\n>0$. Let $C(i,j),i,j\\in[n]$ be the associated $n\\times n$ cost matrix where $C(i,j)=\\infty$ if $(i,j)\\notin E"},"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":"2505.21645","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2025-05-27T18:20:08Z","cross_cats_sorted":["cs.DM"],"title_canon_sha256":"3c25e1d02f82c9014ea58f59a0b3aadb505e3417b87dc76423f8dd63583effdb","abstract_canon_sha256":"3aea31b4c0658eba6327dc14affaac768b2a6b815cad1099689d2e640ea64e62"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:11:01.491718Z","signature_b64":"nBzNBj1E5fE9xO8EUXyMVyNhV/psFe+o7HAKLlIMYmMC24HtExn6V1qa9GVRr6SBJEw8XnaAB7JGOXOHJ+0bBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"59f2bdbca8135a4cdab2fb8527f65b932590cf619ef7586ad4a57390f511f18d","last_reissued_at":"2026-07-05T11:11:01.491156Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:11:01.491156Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Karp's patching algorithm on dense digraph","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"Alan Frieze","submitted_at":"2025-05-27T18:20:08Z","abstract_excerpt":"We consider the following question. We are given a dense digraph $D$ with $n$ vertices and minimum in- and out-degree at least $\\alpha n$, where $\\alpha>1/2$ is a constant. The edges $E(D)$ of $D$ are given independent edge costs $C(e),e\\in E(D)$, such that (i) $C$ has a density $f$ that satisfies $f(x)=a+bx+O(x^2)$, for constants $a>0,b$ as $x\\to 0$ and such that in general either (ii) $\\Pr(C\\geq x)\\leq \\a e^{-\\b x}$ for constants $\\a,\\b>0$, or $f(x)=0$ for $x>\\n$ for some constant $\\n>0$. Let $C(i,j),i,j\\in[n]$ be the associated $n\\times n$ cost matrix where $C(i,j)=\\infty$ if $(i,j)\\notin E"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2505.21645","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/2505.21645/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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2505.21645","created_at":"2026-07-05T11:11:01.491215+00:00"},{"alias_kind":"arxiv_version","alias_value":"2505.21645v1","created_at":"2026-07-05T11:11:01.491215+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2505.21645","created_at":"2026-07-05T11:11:01.491215+00:00"},{"alias_kind":"pith_short_12","alias_value":"LHZL3PFICNNE","created_at":"2026-07-05T11:11:01.491215+00:00"},{"alias_kind":"pith_short_16","alias_value":"LHZL3PFICNNEZWVS","created_at":"2026-07-05T11:11:01.491215+00:00"},{"alias_kind":"pith_short_8","alias_value":"LHZL3PFI","created_at":"2026-07-05T11:11:01.491215+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/LHZL3PFICNNEZWVS7OCSP5S3SM","json":"https://pith.science/pith/LHZL3PFICNNEZWVS7OCSP5S3SM.json","graph_json":"https://pith.science/api/pith-number/LHZL3PFICNNEZWVS7OCSP5S3SM/graph.json","events_json":"https://pith.science/api/pith-number/LHZL3PFICNNEZWVS7OCSP5S3SM/events.json","paper":"https://pith.science/paper/LHZL3PFI"},"agent_actions":{"view_html":"https://pith.science/pith/LHZL3PFICNNEZWVS7OCSP5S3SM","download_json":"https://pith.science/pith/LHZL3PFICNNEZWVS7OCSP5S3SM.json","view_paper":"https://pith.science/paper/LHZL3PFI","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2505.21645&json=true","fetch_graph":"https://pith.science/api/pith-number/LHZL3PFICNNEZWVS7OCSP5S3SM/graph.json","fetch_events":"https://pith.science/api/pith-number/LHZL3PFICNNEZWVS7OCSP5S3SM/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/LHZL3PFICNNEZWVS7OCSP5S3SM/action/timestamp_anchor","attest_storage":"https://pith.science/pith/LHZL3PFICNNEZWVS7OCSP5S3SM/action/storage_attestation","attest_author":"https://pith.science/pith/LHZL3PFICNNEZWVS7OCSP5S3SM/action/author_attestation","sign_citation":"https://pith.science/pith/LHZL3PFICNNEZWVS7OCSP5S3SM/action/citation_signature","submit_replication":"https://pith.science/pith/LHZL3PFICNNEZWVS7OCSP5S3SM/action/replication_record"}},"created_at":"2026-07-05T11:11:01.491215+00:00","updated_at":"2026-07-05T11:11:01.491215+00:00"}