{"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"}