{"id":"https://openalex.org/W2135756775","doi":"https://doi.org/10.1017/s0963548300000675","title":"A Mildly Exponential Time Algorithm for Approximating the Number of Solutions to a Multidimensional Knapsack Problem","display_name":"A Mildly Exponential Time Algorithm for Approximating the Number of Solutions to a Multidimensional Knapsack Problem","publication_year":1993,"publication_date":"1993-09-01","ids":{"openalex":"https://openalex.org/W2135756775","doi":"https://doi.org/10.1017/s0963548300000675","mag":"2135756775"},"language":"en","primary_location":{"id":"doi:10.1017/s0963548300000675","is_oa":false,"landing_page_url":"https://doi.org/10.1017/s0963548300000675","pdf_url":null,"source":{"id":"https://openalex.org/S7291079","display_name":"Combinatorics Probability Computing","issn_l":"0963-5483","issn":["0963-5483","1469-2163"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310311721","host_organization_name":"Cambridge University Press","host_organization_lineage":["https://openalex.org/P4310311721","https://openalex.org/P4310311702"],"host_organization_lineage_names":["Cambridge University Press","University of Cambridge"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Combinatorics, Probability and Computing","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5019297501","display_name":"Martin Dyer","orcid":"https://orcid.org/0000-0002-2018-0374"},"institutions":[{"id":"https://openalex.org/I130828816","display_name":"University of Leeds","ror":"https://ror.org/024mrxd33","country_code":"GB","type":"education","lineage":["https://openalex.org/I130828816"]}],"countries":["GB"],"is_corresponding":true,"raw_author_name":"Martin Dyer","raw_affiliation_strings":["University of Leeds, Leeds LS2 9JT, UK","University of Leeds, Leeds LS2 9JT, UK#TAB#"],"affiliations":[{"raw_affiliation_string":"University of Leeds, Leeds LS2 9JT, UK","institution_ids":["https://openalex.org/I130828816"]},{"raw_affiliation_string":"University of Leeds, Leeds LS2 9JT, UK#TAB#","institution_ids":["https://openalex.org/I130828816"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5056950019","display_name":"Alan Frieze","orcid":"https://orcid.org/0000-0002-8481-5615"},"institutions":[{"id":"https://openalex.org/I74973139","display_name":"Carnegie Mellon University","ror":"https://ror.org/05x2bcf33","country_code":"US","type":"education","lineage":["https://openalex.org/I74973139"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Alan Frieze","raw_affiliation_strings":["Carnegie Mellon University, Pittsburgh PA15213, USA"],"affiliations":[{"raw_affiliation_string":"Carnegie Mellon University, Pittsburgh PA15213, USA","institution_ids":["https://openalex.org/I74973139"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5059436381","display_name":"Ravi Kannan","orcid":null},"institutions":[{"id":"https://openalex.org/I74973139","display_name":"Carnegie Mellon University","ror":"https://ror.org/05x2bcf33","country_code":"US","type":"education","lineage":["https://openalex.org/I74973139"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Ravi Kannan","raw_affiliation_strings":["Carnegie Mellon University, Pittsburgh PA15213, USA"],"affiliations":[{"raw_affiliation_string":"Carnegie Mellon University, Pittsburgh PA15213, USA","institution_ids":["https://openalex.org/I74973139"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5027416299","display_name":"Ajai Kapoor","orcid":null},"institutions":[{"id":"https://openalex.org/I74973139","display_name":"Carnegie Mellon University","ror":"https://ror.org/05x2bcf33","country_code":"US","type":"education","lineage":["https://openalex.org/I74973139"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Ajai Kapoor","raw_affiliation_strings":["Carnegie Mellon University, Pittsburgh PA15213, USA"],"affiliations":[{"raw_affiliation_string":"Carnegie Mellon University, Pittsburgh PA15213, USA","institution_ids":["https://openalex.org/I74973139"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5049775039","display_name":"Ljubomir Perkovi\u0107","orcid":null},"institutions":[{"id":"https://openalex.org/I74973139","display_name":"Carnegie Mellon University","ror":"https://ror.org/05x2bcf33","country_code":"US","type":"education","lineage":["https://openalex.org/I74973139"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Ljubomir Perkovic","raw_affiliation_strings":["Carnegie Mellon University, Pittsburgh PA15213, USA"],"affiliations":[{"raw_affiliation_string":"Carnegie Mellon University, Pittsburgh PA15213, USA","institution_ids":["https://openalex.org/I74973139"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5071039750","display_name":"Umesh Vazirani","orcid":null},"institutions":[{"id":"https://openalex.org/I95457486","display_name":"University of California, Berkeley","ror":"https://ror.org/01an7q238","country_code":"US","type":"education","lineage":["https://openalex.org/I95457486"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Umesh Vazirani","raw_affiliation_strings":["University of California, Berkeley CA94320, USA"],"affiliations":[{"raw_affiliation_string":"University of California, Berkeley CA94320, USA","institution_ids":["https://openalex.org/I95457486"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":6,"corresponding_author_ids":["https://openalex.org/A5019297501"],"corresponding_institution_ids":["https://openalex.org/I130828816"],"apc_list":null,"apc_paid":null,"fwci":1.9203,"has_fulltext":false,"cited_by_count":53,"citation_normalized_percentile":{"value":0.87708012,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":98},"biblio":{"volume":"2","issue":"3","first_page":"271","last_page":"284"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12288","display_name":"Optimization and Search Problems","score":0.9993000030517578,"subfield":{"id":"https://openalex.org/subfields/1705","display_name":"Computer Networks and Communications"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T12288","display_name":"Optimization and Search Problems","score":0.9993000030517578,"subfield":{"id":"https://openalex.org/subfields/1705","display_name":"Computer Networks and Communications"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T12176","display_name":"Optimization and Packing Problems","score":0.9980000257492065,"subfield":{"id":"https://openalex.org/subfields/2209","display_name":"Industrial and Manufacturing Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10996","display_name":"Computational Geometry and Mesh Generation","score":0.9947999715805054,"subfield":{"id":"https://openalex.org/subfields/1704","display_name":"Computer Graphics and Computer-Aided Design"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/knapsack-problem","display_name":"Knapsack problem","score":0.9534683227539062},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6790194511413574},{"id":"https://openalex.org/keywords/markov-chain","display_name":"Markov chain","score":0.6520092487335205},{"id":"https://openalex.org/keywords/change-making-problem","display_name":"Change-making problem","score":0.6318920254707336},{"id":"https://openalex.org/keywords/continuous-knapsack-problem","display_name":"Continuous knapsack problem","score":0.6235136389732361},{"id":"https://openalex.org/keywords/integer","display_name":"Integer (computer science)","score":0.5665991902351379},{"id":"https://openalex.org/keywords/exponential-function","display_name":"Exponential function","score":0.5616867542266846},{"id":"https://openalex.org/keywords/randomized-algorithm","display_name":"Randomized algorithm","score":0.4632478952407837},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.4419933557510376},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.3805437982082367},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.35134583711624146},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.3457241356372833},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.21776026487350464},{"id":"https://openalex.org/keywords/statistics","display_name":"Statistics","score":0.14244672656059265}],"concepts":[{"id":"https://openalex.org/C113138325","wikidata":"https://www.wikidata.org/wiki/Q864457","display_name":"Knapsack problem","level":2,"score":0.9534683227539062},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6790194511413574},{"id":"https://openalex.org/C98763669","wikidata":"https://www.wikidata.org/wiki/Q176645","display_name":"Markov chain","level":2,"score":0.6520092487335205},{"id":"https://openalex.org/C17722475","wikidata":"https://www.wikidata.org/wiki/Q3406279","display_name":"Change-making problem","level":4,"score":0.6318920254707336},{"id":"https://openalex.org/C94569963","wikidata":"https://www.wikidata.org/wiki/Q5165487","display_name":"Continuous knapsack problem","level":3,"score":0.6235136389732361},{"id":"https://openalex.org/C97137487","wikidata":"https://www.wikidata.org/wiki/Q729138","display_name":"Integer (computer science)","level":2,"score":0.5665991902351379},{"id":"https://openalex.org/C151376022","wikidata":"https://www.wikidata.org/wiki/Q168698","display_name":"Exponential function","level":2,"score":0.5616867542266846},{"id":"https://openalex.org/C128669082","wikidata":"https://www.wikidata.org/wiki/Q583461","display_name":"Randomized algorithm","level":2,"score":0.4632478952407837},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.4419933557510376},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.3805437982082367},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.35134583711624146},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.3457241356372833},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.21776026487350464},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.14244672656059265},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.0},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1017/s0963548300000675","is_oa":false,"landing_page_url":"https://doi.org/10.1017/s0963548300000675","pdf_url":null,"source":{"id":"https://openalex.org/S7291079","display_name":"Combinatorics Probability Computing","issn_l":"0963-5483","issn":["0963-5483","1469-2163"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310311721","host_organization_name":"Cambridge University Press","host_organization_lineage":["https://openalex.org/P4310311721","https://openalex.org/P4310311702"],"host_organization_lineage_names":["Cambridge University Press","University of Cambridge"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Combinatorics, Probability and Computing","raw_type":"journal-article"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.421.5556","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.421.5556","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.math.cmu.edu/~af1p/Texfiles/knapsack.pdf","raw_type":"text"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":21,"referenced_works":["https://openalex.org/W1507910450","https://openalex.org/W1559116151","https://openalex.org/W1569752386","https://openalex.org/W1594205593","https://openalex.org/W1999067727","https://openalex.org/W2002234069","https://openalex.org/W2024710200","https://openalex.org/W2046097442","https://openalex.org/W2051283034","https://openalex.org/W2063342497","https://openalex.org/W2069691149","https://openalex.org/W2072211488","https://openalex.org/W2103012681","https://openalex.org/W2103118679","https://openalex.org/W2106285343","https://openalex.org/W2159000620","https://openalex.org/W3111890340","https://openalex.org/W4214936872","https://openalex.org/W4233413206","https://openalex.org/W4298414065","https://openalex.org/W6667732605"],"related_works":["https://openalex.org/W2005493932","https://openalex.org/W2889442766","https://openalex.org/W2168163332","https://openalex.org/W2378557713","https://openalex.org/W2412614002","https://openalex.org/W42170116","https://openalex.org/W2048876360","https://openalex.org/W2009443800","https://openalex.org/W1969981152","https://openalex.org/W2087209118"],"abstract_inverted_index":{"We":[0],"describe":[1],"a":[2,14,44],"time":[3],"randomized":[4],"algorithm":[5,42],"that":[6],"estimates":[7],"the":[8,23,29,36,55],"number":[9,30,37],"of":[10,13,22,31,38],"feasible":[11],"solutions":[12],"multidimensional":[15],"knapsack":[16],"problem":[17],"within":[18],"1":[19],"\u00b1":[20],"\u03b5":[21],"exact":[24],"number.":[25],"(Here":[26],"r":[27],"is":[28,35],"constraints":[32],"and":[33],"n":[34],"integer":[39],"variables.)":[40],"The":[41],"uses":[43],"Markov":[45],"chain":[46],"to":[47,54],"generate":[48],"an":[49],"almost":[50],"uniform":[51],"random":[52],"solution":[53],"problem.":[56]},"counts_by_year":[{"year":2024,"cited_by_count":1},{"year":2023,"cited_by_count":2},{"year":2022,"cited_by_count":3},{"year":2020,"cited_by_count":1},{"year":2019,"cited_by_count":2},{"year":2016,"cited_by_count":4},{"year":2015,"cited_by_count":1},{"year":2014,"cited_by_count":4},{"year":2013,"cited_by_count":3},{"year":2012,"cited_by_count":2}],"updated_date":"2026-04-05T17:49:38.594831","created_date":"2025-10-10T00:00:00"}
