{"id":"https://openalex.org/W2282655154","doi":"https://doi.org/10.1007/s00453-016-0116-0","title":"Approximating Vector Scheduling: Almost Matching Upper and Lower Bounds","display_name":"Approximating Vector Scheduling: Almost Matching Upper and Lower Bounds","publication_year":2016,"publication_date":"2016-01-19","ids":{"openalex":"https://openalex.org/W2282655154","doi":"https://doi.org/10.1007/s00453-016-0116-0","mag":"2282655154","pmid":"https://pubmed.ncbi.nlm.nih.gov/32355383"},"language":"en","primary_location":{"id":"doi:10.1007/s00453-016-0116-0","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s00453-016-0116-0","pdf_url":"https://link.springer.com/content/pdf/10.1007%2Fs00453-016-0116-0.pdf","source":{"id":"https://openalex.org/S89324355","display_name":"Algorithmica","issn_l":"0178-4617","issn":["0178-4617","1432-0541"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Algorithmica","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","pubmed"],"open_access":{"is_oa":true,"oa_status":"hybrid","oa_url":"https://link.springer.com/content/pdf/10.1007%2Fs00453-016-0116-0.pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5008689083","display_name":"Nikhil Bansal","orcid":"https://orcid.org/0000-0002-6290-0894"},"institutions":[{"id":"https://openalex.org/I83019370","display_name":"Eindhoven University of Technology","ror":"https://ror.org/02c2kyt77","country_code":"NL","type":"education","lineage":["https://openalex.org/I83019370"]}],"countries":["NL"],"is_corresponding":true,"raw_author_name":"Nikhil Bansal","raw_affiliation_strings":["1Eindhoven University of Technology, Eindhoven, The Netherlands","Eindhoven University of Technology, Eindhoven, The Netherlands"],"affiliations":[{"raw_affiliation_string":"1Eindhoven University of Technology, Eindhoven, The Netherlands","institution_ids":["https://openalex.org/I83019370"]},{"raw_affiliation_string":"Eindhoven University of Technology, Eindhoven, The Netherlands","institution_ids":["https://openalex.org/I83019370"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5086146785","display_name":"Tim Oosterwijk","orcid":"https://orcid.org/0000-0002-8509-7002"},"institutions":[{"id":"https://openalex.org/I34352273","display_name":"Maastricht University","ror":"https://ror.org/02jz4aj89","country_code":"NL","type":"education","lineage":["https://openalex.org/I34352273"]}],"countries":["NL"],"is_corresponding":false,"raw_author_name":"Tim Oosterwijk","raw_affiliation_strings":["2Maastricht University, Maastricht, The Netherlands","Maastricht University, Maastricht, The Netherlands"],"affiliations":[{"raw_affiliation_string":"2Maastricht University, Maastricht, The Netherlands","institution_ids":["https://openalex.org/I34352273"]},{"raw_affiliation_string":"Maastricht University, Maastricht, The Netherlands","institution_ids":["https://openalex.org/I34352273"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5034888963","display_name":"Tjark Vredeveld","orcid":"https://orcid.org/0000-0002-6357-8622"},"institutions":[{"id":"https://openalex.org/I34352273","display_name":"Maastricht University","ror":"https://ror.org/02jz4aj89","country_code":"NL","type":"education","lineage":["https://openalex.org/I34352273"]}],"countries":["NL"],"is_corresponding":false,"raw_author_name":"Tjark Vredeveld","raw_affiliation_strings":["2Maastricht University, Maastricht, The Netherlands","Maastricht University, Maastricht, The Netherlands"],"affiliations":[{"raw_affiliation_string":"2Maastricht University, Maastricht, The Netherlands","institution_ids":["https://openalex.org/I34352273"]},{"raw_affiliation_string":"Maastricht University, Maastricht, The Netherlands","institution_ids":["https://openalex.org/I34352273"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5041229131","display_name":"Ruben van der Zwaan","orcid":null},"institutions":[{"id":"https://openalex.org/I83019370","display_name":"Eindhoven University of Technology","ror":"https://ror.org/02c2kyt77","country_code":"NL","type":"education","lineage":["https://openalex.org/I83019370"]}],"countries":["NL"],"is_corresponding":false,"raw_author_name":"Ruben van der Zwaan","raw_affiliation_strings":["1Eindhoven University of Technology, Eindhoven, The Netherlands","Eindhoven University of Technology, Eindhoven, The Netherlands"],"affiliations":[{"raw_affiliation_string":"1Eindhoven University of Technology, Eindhoven, The Netherlands","institution_ids":["https://openalex.org/I83019370"]},{"raw_affiliation_string":"Eindhoven University of Technology, Eindhoven, The Netherlands","institution_ids":["https://openalex.org/I83019370"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":4,"corresponding_author_ids":["https://openalex.org/A5008689083"],"corresponding_institution_ids":["https://openalex.org/I83019370"],"apc_list":{"value":2290,"currency":"EUR","value_usd":2890},"apc_paid":{"value":2290,"currency":"EUR","value_usd":2890},"fwci":1.6982,"has_fulltext":true,"cited_by_count":19,"citation_normalized_percentile":{"value":0.85391572,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":98},"biblio":{"volume":"76","issue":"4","first_page":"1077","last_page":"1096"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10551","display_name":"Scheduling and Optimization Algorithms","score":0.9987999796867371,"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"}},"topics":[{"id":"https://openalex.org/T10551","display_name":"Scheduling and Optimization Algorithms","score":0.9987999796867371,"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/T12288","display_name":"Optimization and Search Problems","score":0.9986000061035156,"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/T10720","display_name":"Complexity and Algorithms in Graphs","score":0.9973999857902527,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"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/theory-of-computation","display_name":"Theory of computation","score":0.6967213749885559},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.5775074362754822},{"id":"https://openalex.org/keywords/upper-and-lower-bounds","display_name":"Upper and lower bounds","score":0.5464873909950256},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.5424559712409973},{"id":"https://openalex.org/keywords/scheduling","display_name":"Scheduling (production processes)","score":0.4915355145931244},{"id":"https://openalex.org/keywords/matching","display_name":"Matching (statistics)","score":0.4913858473300934},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.3779861032962799},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.37522032856941223},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.33108237385749817},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.2957320213317871},{"id":"https://openalex.org/keywords/statistics","display_name":"Statistics","score":0.14263588190078735}],"concepts":[{"id":"https://openalex.org/C24858836","wikidata":"https://www.wikidata.org/wiki/Q844718","display_name":"Theory of computation","level":2,"score":0.6967213749885559},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.5775074362754822},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.5464873909950256},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5424559712409973},{"id":"https://openalex.org/C206729178","wikidata":"https://www.wikidata.org/wiki/Q2271896","display_name":"Scheduling (production processes)","level":2,"score":0.4915355145931244},{"id":"https://openalex.org/C165064840","wikidata":"https://www.wikidata.org/wiki/Q1321061","display_name":"Matching (statistics)","level":2,"score":0.4913858473300934},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.3779861032962799},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.37522032856941223},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.33108237385749817},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.2957320213317871},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.14263588190078735},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.0}],"mesh":[],"locations_count":6,"locations":[{"id":"doi:10.1007/s00453-016-0116-0","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s00453-016-0116-0","pdf_url":"https://link.springer.com/content/pdf/10.1007%2Fs00453-016-0116-0.pdf","source":{"id":"https://openalex.org/S89324355","display_name":"Algorithmica","issn_l":"0178-4617","issn":["0178-4617","1432-0541"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Algorithmica","raw_type":"journal-article"},{"id":"pmid:32355383","is_oa":false,"landing_page_url":"https://pubmed.ncbi.nlm.nih.gov/32355383","pdf_url":null,"source":{"id":"https://openalex.org/S4306525036","display_name":"PubMed","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1299303238","host_organization_name":"National Institutes of Health","host_organization_lineage":["https://openalex.org/I1299303238"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Algorithmica","raw_type":null},{"id":"pmh:oai:cris.maastrichtuniversity.nl:openaire/ced6d00e-acc1-42d3-b22b-faeb17dbffaf","is_oa":true,"landing_page_url":"https://cris.maastrichtuniversity.nl/en/publications/ced6d00e-acc1-42d3-b22b-faeb17dbffaf","pdf_url":null,"source":{"id":"https://openalex.org/S4306402616","display_name":"Research Publications (Maastricht University)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I34352273","host_organization_name":"Maastricht University","host_organization_lineage":["https://openalex.org/I34352273"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Bansal, N, Oosterwijk, T, Vredeveld, T & Van Der Zwaan, R 2016, 'Approximating Vector Scheduling: Almost Matching Upper and Lower Bounds', Algorithmica, vol. 76, no. 4, pp. 1077-1096. https://doi.org/10.1007/s00453-016-0116-0","raw_type":"info:eu-repo/semantics/publishedVersion"},{"id":"pmh:oai:pure.tue.nl:openaire/03d9bc9d-38e4-4761-8147-8a20d48a1d2c","is_oa":true,"landing_page_url":"https://research.tue.nl/en/publications/03d9bc9d-38e4-4761-8147-8a20d48a1d2c","pdf_url":null,"source":{"id":"https://openalex.org/S4406922641","display_name":"TU/e Research Portal","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":"other-oa","license_id":"https://openalex.org/licenses/other-oa","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Bansal, N, Oosterwijk, T, Vredeveld, T & van der Zwaan, R 2016, 'Approximating vector scheduling: almost matching upper and lower bounds', Algorithmica, vol. 76, no. 4, pp. 1077-1096. https://doi.org/10.1007/s00453-016-0116-0","raw_type":"info:eu-repo/semantics/publishedVersion"},{"id":"pmh:oai:pubmedcentral.nih.gov:7175655","is_oa":true,"landing_page_url":"https://www.ncbi.nlm.nih.gov/pmc/articles/7175655","pdf_url":null,"source":{"id":"https://openalex.org/S2764455111","display_name":"PubMed Central","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1299303238","host_organization_name":"National Institutes of Health","host_organization_lineage":["https://openalex.org/I1299303238"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Algorithmica","raw_type":"Text"},{"id":"pmh:tue:oai:pure.tue.nl:publications/03d9bc9d-38e4-4761-8147-8a20d48a1d2c","is_oa":true,"landing_page_url":"https://research.tue.nl/nl/publications/03d9bc9d-38e4-4761-8147-8a20d48a1d2c","pdf_url":null,"source":{"id":"https://openalex.org/S4306401843","display_name":"Data Archiving and Networked Services (DANS)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1322597698","host_organization_name":"Royal Netherlands Academy of Arts and Sciences","host_organization_lineage":["https://openalex.org/I1322597698"],"host_organization_lineage_names":[],"type":"repository"},"license":"other-oa","license_id":"https://openalex.org/licenses/other-oa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Algorithmica, 76(4), 1077 - 1096. Springer","raw_type":"info:eu-repo/semantics/article"}],"best_oa_location":{"id":"doi:10.1007/s00453-016-0116-0","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s00453-016-0116-0","pdf_url":"https://link.springer.com/content/pdf/10.1007%2Fs00453-016-0116-0.pdf","source":{"id":"https://openalex.org/S89324355","display_name":"Algorithmica","issn_l":"0178-4617","issn":["0178-4617","1432-0541"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Algorithmica","raw_type":"journal-article"},"sustainable_development_goals":[{"score":0.47999998927116394,"display_name":"Decent work and economic growth","id":"https://metadata.un.org/sdg/8"}],"awards":[{"id":"https://openalex.org/G1664046972","display_name":null,"funder_award_id":"639.022.211","funder_id":"https://openalex.org/F4320321800","funder_display_name":"Nederlandse Organisatie voor Wetenschappelijk Onderzoek"},{"id":"https://openalex.org/G5365867299","display_name":null,"funder_award_id":"Grant","funder_id":"https://openalex.org/F4320321800","funder_display_name":"Nederlandse Organisatie voor Wetenschappelijk Onderzoek"},{"id":"https://openalex.org/G628422989","display_name":null,"funder_award_id":"617951","funder_id":"https://openalex.org/F4320334678","funder_display_name":"European Research Council"}],"funders":[{"id":"https://openalex.org/F4320321800","display_name":"Nederlandse Organisatie voor Wetenschappelijk Onderzoek","ror":"https://ror.org/04jsz6e67"},{"id":"https://openalex.org/F4320334678","display_name":"European Research Council","ror":"https://ror.org/0472cxd90"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2282655154.pdf","grobid_xml":"https://content.openalex.org/works/W2282655154.grobid-xml"},"referenced_works_count":21,"referenced_works":["https://openalex.org/W1595920282","https://openalex.org/W1977276352","https://openalex.org/W1995725694","https://openalex.org/W2001260910","https://openalex.org/W2011039300","https://openalex.org/W2014071862","https://openalex.org/W2026129593","https://openalex.org/W2026405784","https://openalex.org/W2042276676","https://openalex.org/W2071373611","https://openalex.org/W2073271726","https://openalex.org/W2090960684","https://openalex.org/W2093979815","https://openalex.org/W2114616381","https://openalex.org/W2117931860","https://openalex.org/W2123068254","https://openalex.org/W2159892713","https://openalex.org/W2160579444","https://openalex.org/W2964162348","https://openalex.org/W3161056635","https://openalex.org/W4285719527"],"related_works":["https://openalex.org/W1972035260","https://openalex.org/W4250775169","https://openalex.org/W2794488505","https://openalex.org/W4301594054","https://openalex.org/W4309231192","https://openalex.org/W3125889879","https://openalex.org/W2064563243","https://openalex.org/W3124422538","https://openalex.org/W2295467472","https://openalex.org/W2046228374"],"abstract_inverted_index":{"We":[0,289],"consider":[1],"the":[2,10,44,49,55,63,72,79,130,218,332,339],"Vector":[3],"Scheduling":[4],"problem,":[5],"a":[6,143,295],"natural":[7],"generalization":[8],"of":[9,58],"classical":[11],"makespan":[12],"minimization":[13],"problem":[14,73],"to":[15,47,51],"multiple":[16],"resources.":[17],"Here,":[18],"we":[19,140,162],"are":[20,273],"given":[21],"<i>n</i>":[22],"jobs,":[23],"represented":[24],"as":[25],"<i>d</i>-dimensional":[26],"vectors":[27],"in":[28,126,306],"<mml:math":[29,85,96,119,164,171,181,190,201,225,236,267,282,296,308],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">":[30],"<mml:msup><mml:mrow><mml:mo>[</mml:mo>":[31],"<mml:mn>0</mml:mn>":[32],"<mml:mo>,</mml:mo>":[33,90,100],"<mml:mn>1</mml:mn>":[34,105,192,206,227,240,298,312],"<mml:mo>]</mml:mo></mml:mrow>":[35],"<mml:mi>d</mml:mi></mml:msup>":[36],"</mml:math>":[37,94,117,122,169,174,185,196,216,231,251,270,286,302,328],",":[38,186],"and":[39,43,78,151,175],"<i>m</i>":[40],"identical":[41],"machines,":[42],"goal":[45],"is":[46,65,84,134,149,188],"assign":[48],"jobs":[50],"machines":[52,272],"such":[53],"that":[54,142,304],"maximum":[56],"<i>load</i>":[57],"each":[59],"machine":[60],"over":[61],"all":[62],"coordinates":[64],"at":[66],"most":[67],"1.":[68],"For":[69,179],"fixed":[70],"<i>d</i>,":[71],"admits":[74],"an":[75,153],"approximation":[76,335],"scheme,":[77],"best":[80],"known":[81],"running":[82,159,199,234],"time":[83,200,235,257,307],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:msup><mml:mi>n</mml:mi>":[86],"<mml:mrow><mml:mi>f</mml:mi>":[87],"<mml:mo>(</mml:mo>":[88,166,204,245,317],"<mml:mi>\u03f5</mml:mi>":[89,99,107,194,208,229,242,300,314],"<mml:mi>d</mml:mi>":[91,101,112,246,318,321],"<mml:mo>)</mml:mo></mml:mrow>":[92,102,108,113,168,195,230,247,301,315,322],"</mml:msup>":[93,115,213,248,323],"where":[95],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow><mml:mi>f</mml:mi>":[97],"<mml:mrow><mml:mo>(</mml:mo>":[98,111],"<mml:mo>=</mml:mo>":[103],"<mml:msup><mml:mrow><mml:mo>(</mml:mo>":[104,311],"<mml:mo>/</mml:mo>":[106,207,211,241,313],"<mml:mrow><mml:mover><mml:mi>O</mml:mi>":[109],"<mml:mo>~</mml:mo></mml:mover>":[110,121],"</mml:mrow>":[114,116,215,250],"(":[118],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mover><mml:mi>O</mml:mi>":[120],"suppresses":[123],"polylogarithmic":[124],"terms":[125],"<i>d</i>).":[127],"In":[128,137],"particular,":[129],"dependence":[131,146],"on":[132,147],"<i>d</i>":[133,148],"double":[135,144],"exponential.":[136],"this":[138],"paper":[139],"show":[141,176],"exponential":[145],"necessary,":[150],"give":[152],"improved":[154],"algorithm":[155],"with":[156,198,233,276,294],"essentially":[157],"optimal":[158],"time.":[160],"Specifically,":[161],"let":[163],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow><mml:mo>exp</mml:mo>":[165,202,237,309],"<mml:mi>x</mml:mi>":[167],"denote":[170],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:msup><mml:mn>2</mml:mn>":[172],"<mml:mi>x</mml:mi></mml:msup>":[173],"that:":[177],"(1)":[178],"any":[180],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow><mml:mi>\u03f5</mml:mi>":[182,268,283],"<mml:mo><</mml:mo>":[183],"<mml:mn>1</mml:mn></mml:mrow>":[184],"there":[187],"no":[189],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow><mml:mo>(</mml:mo>":[191,226,297],"<mml:mo>+</mml:mo>":[193,228,299,325],"-approximation":[197,232,303],"<mml:mfenced><mml:mi>o</mml:mi>":[203],"<mml:msup><mml:mrow><mml:mo>\u230a</mml:mo>":[205,239],"<mml:mo>\u230b</mml:mo></mml:mrow>":[209,243],"<mml:mrow><mml:mi>d</mml:mi>":[210],"<mml:mn>3</mml:mn></mml:mrow>":[212],"<mml:mo>)</mml:mo></mml:mfenced>":[214],"unless":[217,253],"Exponential":[219],"Time":[220],"Hypothesis":[221],"fails.":[222],"(2)":[223],"No":[224],"<mml:mfenced>":[238,310],"<mml:mrow><mml:mi>o</mml:mi>":[244],"</mml:mfenced>":[249,324],"exists,":[252],"NP":[254],"has":[255],"subexponential":[256],"algorithms.":[258],"(3)":[259],"Similar":[260],"lower":[261,292],"bounds":[262,293],"also":[263],"hold":[264],"even":[265],"if":[266],"<mml:mi>m</mml:mi></mml:mrow>":[269],"extra":[271],"allowed":[274],"(i.e.":[275],"resource":[277],"augmentation),":[278],"for":[279,338],"sufficiently":[280],"small":[281],"<mml:mo>></mml:mo>":[284],"<mml:mn>0</mml:mn></mml:mrow>":[285],".":[287,329],"(4)":[288],"complement":[290],"these":[291],"runs":[305],"<mml:mrow><mml:mi>O</mml:mi>":[316],"<mml:mo>log</mml:mo>":[319,320],"<mml:mi>n</mml:mi>":[326],"<mml:mi>d</mml:mi></mml:mrow>":[327],"This":[330],"gives":[331],"first":[333],"efficient":[334],"scheme":[336],"(EPTAS)":[337],"problem.":[340]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2024,"cited_by_count":2},{"year":2023,"cited_by_count":4},{"year":2021,"cited_by_count":2},{"year":2020,"cited_by_count":4},{"year":2018,"cited_by_count":1},{"year":2017,"cited_by_count":2},{"year":2016,"cited_by_count":1},{"year":2015,"cited_by_count":2}],"updated_date":"2026-04-10T15:06:20.359241","created_date":"2025-10-10T00:00:00"}
