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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 $"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1802.05791","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2018-02-15T23:21:58Z","cross_cats_sorted":[],"title_canon_sha256":"18dd60831a05e2bfbd6c9576aa071ad2ecae0a216c20d5acb2ec5740fe4c7999","abstract_canon_sha256":"c7950d4a8544c11ebcbfe72a830056499d21d2b14c326e5bf17abf15e6523738"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:23:11.014805Z","signature_b64":"XCphohZhuUq9tvJEDs1ybz7eaTbZan1t6bZ1TMTQc4EF5Xk9u2IV0mbXP4HVwoee5bZFecGrcOMspTVHSKMECQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4f4b3c1b9d69706ca820490249aa93af1d53b784bca51769aec63e7d3b835ff5","last_reissued_at":"2026-05-18T00:23:11.014057Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:23:11.014057Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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 $C$. 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