{"paper":{"title":"Approximations of the Densest k-Subhypergraph and Set Union Knapsack problems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"cs.DS","authors_text":"Richard Taylor","submitted_at":"2016-10-17T00:28:12Z","abstract_excerpt":"For any given $\\epsilon>0$ we provide an algorithm for the Densest $k$-Subhypergraph Problem with an approximation ratio of at most $O(n^{\\theta_m+2\\epsilon})$ for $\\theta_m=\\frac{1}{2}m-\\frac{1}{2}-\\frac{1}{2m}$ and run time at most $O(n^{m-2+1/\\epsilon})$, where the hyperedges have at most $m$ vertices. We use this result to give an algorithm for the Set Union Knapsack Problem with an approximation ratio of at most $O(n^{\\alpha_m+\\epsilon})$ for $\\alpha_m=\\frac{2}{3}[m-1-\\frac{2m-2}{m^2+m-1}]$ and run time at most $O(n^{5(m-2)+9/\\epsilon})$, where the subsets have at most $m$ elements. The a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1610.04935","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":""},"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"}