{"paper":{"title":"Tournament Ranking: Duality and Efficiency","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Ge Song, Mengxi Yang, Wenan Zang","submitted_at":"2026-06-30T12:23:11Z","abstract_excerpt":"The feedback arc set problem on tournaments arises in a rich variety of applications, and has been studied extensively in several research fields over the past six decades. It is well known that this problem is $NP$-hard and admits a polynomial-time approximation scheme (PTAS) in general. A tournament $T=(V, A)$ is called cycle Mengerian (CM) if, for every nonnegative integral weight function defined on $A$, the minimum total weight of a feedback arc set is equal to the maximum size of a cycle packing. In 2020 Chen et al. obtained a structural characterization of all CM tournaments; however, t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.31565","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/2606.31565/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"}