{"paper":{"title":"A Faster FPTAS for #Knapsack","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.DS","authors_text":"Liran Markin, Oren Weimann, Pawe{\\l} Gawrychowski","submitted_at":"2018-02-15T23:21:58Z","abstract_excerpt":"Given a set $W = \\{w_1,\\ldots, w_n\\}$ of non-negative integer weights and an integer $C$, the #Knapsack problem asks to count the number of distinct subsets of $W$ whose total weight is at most $C$. In the more general integer version of the problem, the subsets are multisets. That is, we are also given a set $ \\{u_1,\\ldots, u_n\\}$ and we are allowed to take up to $u_i$ items of weight $w_i$.\n  We present a deterministic FPTAS for #Knapsack running in $O(n^{2.5}\\varepsilon^{-1.5}\\log(n \\varepsilon^{-1})\\log (n \\varepsilon))$ time. The previous best deterministic algorithm [FOCS 2011] runs in $"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1802.05791","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"}