{"paper":{"title":"Approximating set multi-covers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CC","math.MG"],"primary_cat":"math.CO","authors_text":"Alexandr Polyanskii, M\\'arton Nasz\\'odi","submitted_at":"2016-08-03T19:12:23Z","abstract_excerpt":"Johnson and Lov\\'asz and Stein proved independently that any hypergraph satisfies $\\tau\\leq (1+\\ln \\Delta)\\tau^{\\ast}$, where $\\tau$ is the transversal number, $\\tau^{\\ast}$ is its fractional version, and $\\Delta$ denotes the maximum degree. We prove $\\tau_f\\leq c \\tau^{\\ast}\\max\\{\\ln \\Delta, f\\}$ for the $f$-fold transversal number $\\tau_f$. Similarly to Johnson, Lov\\'asz and Stein, we also show that this bound can be achieved non-probabilistically, using a greedy algorithm.\n  As a combinatorial application, we prove an estimate on how fast $\\tau_f/f$ converges to $\\tau^{\\ast}$. As a geometri"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1608.01292","kind":"arxiv","version":2},"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"}