{"id":162000,"date":"2011-10-01T00:00:00","date_gmt":"2011-10-01T00:00:00","guid":{"rendered":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/msr-research-item\/an-fptas-for-knapsack-and-related-counting-problems\/"},"modified":"2018-10-16T20:05:52","modified_gmt":"2018-10-17T03:05:52","slug":"an-fptas-for-knapsack-and-related-counting-problems","status":"publish","type":"msr-research-item","link":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/publication\/an-fptas-for-knapsack-and-related-counting-problems\/","title":{"rendered":"An FPTAS for #Knapsack and Related Counting Problems"},"content":{"rendered":"<div class=\"asset-content\">\n<p>Given n elements with non-negative integer weights w1, . . . ,wn and an integer capacity C, we consider the counting version of the classic knapsack problem: find the number of distinct subsets whose weights add up to at most C. We give the first deterministic, fully polynomial-time approximation scheme (FPTAS) for estimating the number of solutions to any knapsack constraint (our estimate has relative error 1 \u00b1 eps). Our algorithm is based on dynamic programming. Previously, randomized polynomial-time approximation schemes (FPRAS) were known first by Morris and Sinclair via Markov chain Monte Carlo techniques, and subsequently by Dyer via dynamic programming and rejection sampling.<\/p>\n<p>In addition, we present a new method for deterministic approximate counting using read-once branching programs. Our approach yields an FPTAS for several other counting problems, including counting solutions for the multidimensional knapsack problem with a constant number of constraints, the general integer knapsack problem, and the contingency tables problem with a constant number of rows.<\/p>\n<\/div>\n<p><!-- .asset-content --><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Given n elements with non-negative integer weights w1, . . . ,wn and an integer capacity C, we consider the counting version of the classic knapsack problem: find the number of distinct subsets whose weights add up to at most C. We give the first deterministic, fully polynomial-time approximation scheme (FPTAS) for estimating the number [&hellip;]<\/p>\n","protected":false},"featured_media":0,"template":"","meta":{"msr-url-field":"","msr-podcast-episode":"","msrModifiedDate":"","msrModifiedDateEnabled":false,"ep_exclude_from_search":false,"_classifai_error":"","msr-author-ordering":null,"msr_publishername":"IEEE","msr_publisher_other":"","msr_booktitle":"","msr_chapter":"","msr_edition":"FOCS'11","msr_editors":"","msr_how_published":"","msr_isbn":"","msr_issue":"","msr_journal":"","msr_number":"","msr_organization":"","msr_pages_string":"","msr_page_range_start":"","msr_page_range_end":"","msr_series":"","msr_volume":"","msr_copyright":"","msr_conference_name":"FOCS'11","msr_doi":"","msr_arxiv_id":"","msr_s2_paper_id":"","msr_mag_id":"","msr_pubmed_id":"","msr_other_authors":"Adam Klivans, Daniel Stefankovic, Santosh Vempala, Eric 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