{"id":"https://openalex.org/W4396823872","doi":"https://doi.org/10.1145/3618260.3649749","title":"Understanding the Cluster Linear Program for Correlation Clustering","display_name":"Understanding the Cluster Linear Program for Correlation Clustering","publication_year":2024,"publication_date":"2024-06-10","ids":{"openalex":"https://openalex.org/W4396823872","doi":"https://doi.org/10.1145/3618260.3649749"},"language":"en","primary_location":{"id":"doi:10.1145/3618260.3649749","is_oa":true,"landing_page_url":"https://doi.org/10.1145/3618260.3649749","pdf_url":"https://dl.acm.org/doi/pdf/10.1145/3618260.3649749","source":null,"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Proceedings of the 56th Annual ACM Symposium on Theory of Computing","raw_type":"proceedings-article"},"type":"conference-paper","indexed_in":["arxiv","crossref"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://dl.acm.org/doi/pdf/10.1145/3618260.3649749","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5030710270","display_name":"Nairen Cao","orcid":"https://orcid.org/0000-0002-4961-763X"},"institutions":[{"id":"https://openalex.org/I103531236","display_name":"Boston College","ror":"https://ror.org/02n2fzt79","country_code":"US","type":"education","lineage":["https://openalex.org/I103531236"]},{"id":"https://openalex.org/I2803044603","display_name":"Brighton Hospital","ror":"https://ror.org/006yjsn14","country_code":"US","type":"healthcare","lineage":["https://openalex.org/I1320343167","https://openalex.org/I2803044603"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Nairen Cao","raw_affiliation_strings":["Boston College, Brighton, USA"],"raw_orcid":"https://orcid.org/0000-0002-4961-763X","affiliations":[{"raw_affiliation_string":"Boston College, Brighton, USA","institution_ids":["https://openalex.org/I2803044603","https://openalex.org/I103531236"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5074251728","display_name":"Vincent Cohen-Addad","orcid":"https://orcid.org/0000-0002-0779-8962"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Vincent Cohen-Addad","raw_affiliation_strings":["Google Research, Grenoble, France"],"raw_orcid":"https://orcid.org/0000-0002-0779-8962","affiliations":[{"raw_affiliation_string":"Google Research, Grenoble, France","institution_ids":[]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5075423314","display_name":"Euiwoong Lee","orcid":"https://orcid.org/0000-0003-1454-7587"},"institutions":[{"id":"https://openalex.org/I27837315","display_name":"University of Michigan","ror":"https://ror.org/00jmfr291","country_code":"US","type":"education","lineage":["https://openalex.org/I27837315"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Euiwoong Lee","raw_affiliation_strings":["University of Michigan, Ann Arbor, USA"],"raw_orcid":"https://orcid.org/0000-0003-1454-7587","affiliations":[{"raw_affiliation_string":"University of Michigan, Ann Arbor, USA","institution_ids":["https://openalex.org/I27837315"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5100430766","display_name":"Shi Li","orcid":"https://orcid.org/0000-0001-9140-9415"},"institutions":[{"id":"https://openalex.org/I881766915","display_name":"Nanjing University","ror":"https://ror.org/01rxvg760","country_code":"CN","type":"education","lineage":["https://openalex.org/I881766915"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Shi Li","raw_affiliation_strings":["Nanjing University, Nanjing, China"],"raw_orcid":"https://orcid.org/0000-0001-9140-9415","affiliations":[{"raw_affiliation_string":"Nanjing University, Nanjing, China","institution_ids":["https://openalex.org/I881766915"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5056131799","display_name":"Alantha Newman","orcid":"https://orcid.org/0009-0009-7353-7734"},"institutions":[{"id":"https://openalex.org/I1294671590","display_name":"Centre National de la Recherche Scientifique","ror":"https://ror.org/02feahw73","country_code":"FR","type":"government","lineage":["https://openalex.org/I1294671590"]},{"id":"https://openalex.org/I899635006","display_name":"Universit\u00e9 Grenoble Alpes","ror":"https://ror.org/02rx3b187","country_code":"FR","type":"education","lineage":["https://openalex.org/I899635006"]}],"countries":["FR"],"is_corresponding":false,"raw_author_name":"Alantha Newman","raw_affiliation_strings":["CNRS - Universit\u00e9 Grenoble Alpes, Grenoble, France"],"raw_orcid":"https://orcid.org/0009-0009-7353-7734","affiliations":[{"raw_affiliation_string":"CNRS - Universit\u00e9 Grenoble Alpes, Grenoble, France","institution_ids":["https://openalex.org/I899635006","https://openalex.org/I1294671590"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5094131872","display_name":"Lukas Vogl","orcid":"https://orcid.org/0000-0002-8241-536X"},"institutions":[{"id":"https://openalex.org/I5124864","display_name":"\u00c9cole Polytechnique F\u00e9d\u00e9rale de Lausanne","ror":"https://ror.org/02s376052","country_code":"CH","type":"education","lineage":["https://openalex.org/I2799323385","https://openalex.org/I5124864"]}],"countries":["CH"],"is_corresponding":false,"raw_author_name":"Lukas Vogl","raw_affiliation_strings":["\u00c9cole polytechnique f\u00e9d\u00e9rale de Lausanne, Lausanne, Switzerland"],"raw_orcid":"https://orcid.org/0000-0002-8241-536X","affiliations":[{"raw_affiliation_string":"\u00c9cole polytechnique f\u00e9d\u00e9rale de Lausanne, Lausanne, Switzerland","institution_ids":["https://openalex.org/I5124864"]}]}],"institutions":[],"countries_distinct_count":4,"institutions_distinct_count":7,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":10,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":"1605","last_page":"1616"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10720","display_name":"Complexity and Algorithms in Graphs","score":0.9994000196456909,"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"}},"topics":[{"id":"https://openalex.org/T10720","display_name":"Complexity and Algorithms in Graphs","score":0.9994000196456909,"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"}},{"id":"https://openalex.org/T12072","display_name":"Machine Learning and Algorithms","score":0.9955999851226807,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"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/T11106","display_name":"Data Management and Algorithms","score":0.9947999715805054,"subfield":{"id":"https://openalex.org/subfields/1711","display_name":"Signal Processing"},"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/rounding","display_name":"Rounding","score":0.7879072427749634},{"id":"https://openalex.org/keywords/approximation-algorithm","display_name":"Approximation algorithm","score":0.7123380899429321},{"id":"https://openalex.org/keywords/cluster-analysis","display_name":"Cluster analysis","score":0.6713529229164124},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6586941480636597},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.5874377489089966},{"id":"https://openalex.org/keywords/partition","display_name":"Partition (number theory)","score":0.5327650904655457},{"id":"https://openalex.org/keywords/mathematical-proof","display_name":"Mathematical proof","score":0.4760946035385132},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.45230889320373535},{"id":"https://openalex.org/keywords/cluster","display_name":"Cluster (spacecraft)","score":0.45072436332702637},{"id":"https://openalex.org/keywords/theory-of-computation","display_name":"Theory of computation","score":0.44996872544288635},{"id":"https://openalex.org/keywords/simple","display_name":"Simple (philosophy)","score":0.42155298590660095},{"id":"https://openalex.org/keywords/upper-and-lower-bounds","display_name":"Upper and lower bounds","score":0.41533416509628296},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.28975173830986023},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.24206247925758362}],"concepts":[{"id":"https://openalex.org/C136625980","wikidata":"https://www.wikidata.org/wiki/Q663208","display_name":"Rounding","level":2,"score":0.7879072427749634},{"id":"https://openalex.org/C148764684","wikidata":"https://www.wikidata.org/wiki/Q621751","display_name":"Approximation algorithm","level":2,"score":0.7123380899429321},{"id":"https://openalex.org/C73555534","wikidata":"https://www.wikidata.org/wiki/Q622825","display_name":"Cluster analysis","level":2,"score":0.6713529229164124},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6586941480636597},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.5874377489089966},{"id":"https://openalex.org/C42812","wikidata":"https://www.wikidata.org/wiki/Q1082910","display_name":"Partition (number theory)","level":2,"score":0.5327650904655457},{"id":"https://openalex.org/C108710211","wikidata":"https://www.wikidata.org/wiki/Q11538","display_name":"Mathematical proof","level":2,"score":0.4760946035385132},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.45230889320373535},{"id":"https://openalex.org/C164866538","wikidata":"https://www.wikidata.org/wiki/Q367351","display_name":"Cluster (spacecraft)","level":2,"score":0.45072436332702637},{"id":"https://openalex.org/C24858836","wikidata":"https://www.wikidata.org/wiki/Q844718","display_name":"Theory of computation","level":2,"score":0.44996872544288635},{"id":"https://openalex.org/C2780586882","wikidata":"https://www.wikidata.org/wiki/Q7520643","display_name":"Simple (philosophy)","level":2,"score":0.42155298590660095},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.41533416509628296},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.28975173830986023},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.24206247925758362},{"id":"https://openalex.org/C111919701","wikidata":"https://www.wikidata.org/wiki/Q9135","display_name":"Operating system","level":1,"score":0.0},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.0},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.0},{"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/C138885662","wikidata":"https://www.wikidata.org/wiki/Q5891","display_name":"Philosophy","level":0,"score":0.0},{"id":"https://openalex.org/C111472728","wikidata":"https://www.wikidata.org/wiki/Q9471","display_name":"Epistemology","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":3,"locations":[{"id":"doi:10.1145/3618260.3649749","is_oa":true,"landing_page_url":"https://doi.org/10.1145/3618260.3649749","pdf_url":"https://dl.acm.org/doi/pdf/10.1145/3618260.3649749","source":null,"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Proceedings of the 56th Annual ACM Symposium on Theory of Computing","raw_type":"proceedings-article"},{"id":"pmh:oai:arXiv.org:2404.17509","is_oa":true,"landing_page_url":"http://arxiv.org/abs/2404.17509","pdf_url":"https://arxiv.org/pdf/2404.17509","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"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":null,"raw_type":"text"},{"id":"pmh:oai:HAL:hal-04789267v1","is_oa":false,"landing_page_url":"https://hal.science/hal-04789267","pdf_url":null,"source":{"id":"https://openalex.org/S4306402512","display_name":"HAL (Le Centre pour la Communication Scientifique Directe)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1294671590","host_organization_name":"Centre National de la Recherche Scientifique","host_organization_lineage":["https://openalex.org/I1294671590"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"STOC '24: 56th Annual ACM Symposium on Theory of Computing, Jun 2024, Vancouver BC Canada, France. pp.1605-1616, &#x27E8;10.1145/3618260.3649749&#x27E9;","raw_type":"info:eu-repo/semantics/conferenceObject"}],"best_oa_location":{"id":"doi:10.1145/3618260.3649749","is_oa":true,"landing_page_url":"https://doi.org/10.1145/3618260.3649749","pdf_url":"https://dl.acm.org/doi/pdf/10.1145/3618260.3649749","source":null,"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Proceedings of the 56th Annual ACM Symposium on Theory of Computing","raw_type":"proceedings-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G201645195","display_name":null,"funder_award_id":"CCF-2008422","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G3052677248","display_name":null,"funder_award_id":"200021-184656","funder_id":"https://openalex.org/F4320320924","funder_display_name":"Schweizerischer Nationalfonds zur F\u00f6rderung der Wissenschaftlichen Forschung"},{"id":"https://openalex.org/G4915951725","display_name":"Randomness in Problem Instances and Randomized Algorithms","funder_award_id":"184656","funder_id":"https://openalex.org/F4320320924","funder_display_name":"Schweizerischer Nationalfonds zur F\u00f6rderung der Wissenschaftlichen Forschung"},{"id":"https://openalex.org/G5256785375","display_name":null,"funder_award_id":"200021-","funder_id":"https://openalex.org/F4320320924","funder_display_name":"Schweizerischer Nationalfonds zur F\u00f6rderung der Wissenschaftlichen Forschung"},{"id":"https://openalex.org/G7385471316","display_name":"CAREER: Towards Optimal Approximabilities for Clustering","funder_award_id":"2236669","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G7995651076","display_name":"AF: Small: Combinatorial Optimization Problems in Vertex Centric Computation","funder_award_id":"2008422","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G993971353","display_name":null,"funder_award_id":"200021","funder_id":"https://openalex.org/F4320320924","funder_display_name":"Schweizerischer Nationalfonds zur F\u00f6rderung der Wissenschaftlichen Forschung"}],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"},{"id":"https://openalex.org/F4320320924","display_name":"Schweizerischer Nationalfonds zur F\u00f6rderung der Wissenschaftlichen Forschung","ror":"https://ror.org/00yjd3n13"},{"id":"https://openalex.org/F4320324852","display_name":"Nanjing University","ror":"https://ror.org/01rxvg760"}],"has_content":{"pdf":true,"grobid_xml":false},"content_urls":{"pdf":"https://content.openalex.org/works/W4396823872.pdf"},"referenced_works_count":31,"referenced_works":["https://openalex.org/W398859631","https://openalex.org/W1836916015","https://openalex.org/W1978593916","https://openalex.org/W1978658266","https://openalex.org/W1985875030","https://openalex.org/W1999032440","https://openalex.org/W2020254000","https://openalex.org/W2035836267","https://openalex.org/W2083858754","https://openalex.org/W2087187685","https://openalex.org/W2091858563","https://openalex.org/W2117654132","https://openalex.org/W2146081992","https://openalex.org/W2161719898","https://openalex.org/W2164625277","https://openalex.org/W2171179180","https://openalex.org/W2560674852","https://openalex.org/W2949580236","https://openalex.org/W2951430799","https://openalex.org/W3104252707","https://openalex.org/W3121437753","https://openalex.org/W4206566163","https://openalex.org/W4234897719","https://openalex.org/W4289255343","https://openalex.org/W4309237947","https://openalex.org/W4313227193","https://openalex.org/W4316652477","https://openalex.org/W4389615664","https://openalex.org/W4390098347","https://openalex.org/W4390575673","https://openalex.org/W4390575882"],"related_works":["https://openalex.org/W4220780102","https://openalex.org/W2410881844","https://openalex.org/W3196334750","https://openalex.org/W1502401885","https://openalex.org/W2004257129","https://openalex.org/W2116281088","https://openalex.org/W2357551824","https://openalex.org/W2016668641","https://openalex.org/W2032550399","https://openalex.org/W2057791178"],"abstract_inverted_index":{"In":[0,56,167],"the":[1,14,29,37,41,44,49,52,75,89,101,120,138,144,189,201,209,213,236,255,259,262,281,300,303,321],"classic":[2],"Correlation":[3,108,177,204],"Clustering":[4,178],"problem":[5],"introduced":[6],"by":[7,69,137,265,296],"Bansal,":[8],"Blum,":[9],"and":[10,28,62,78,82,162,173,271,280],"Chawla":[11],"\u200d(FOCS":[12,84],"2002),":[13],"input":[15],"is":[16,31,112,126,216],"a":[17,34,67,71,98,171,193,267,278,284,289,307,343],"complete":[18],"graph":[19],"where":[20],"edges":[21],"are":[22,250],"labeled":[23],"either":[24],"+":[25],"or":[26],"\u2212,":[27],"goal":[30],"to":[32,148,169,180,287,298],"find":[33],"partition":[35],"of":[36,43,51,74,92,143,160,203,261,302,318,340],"vertices":[38],"that":[39,197,221],"minimizes":[40],"sum":[42,50],"+edges":[45],"across":[46],"parts":[47],"plus":[48],"-edges":[53],"within":[54],"parts.":[55],"recent":[57],"years,":[58],"Chawla,":[59],"Makarychev,":[60],"Schramm":[61],"Yaroslavtsev":[63],"\u200d(STOC":[64],"2015)":[65],"gave":[66],"2.06-approximation":[68],"providing":[70,235,272],"near-optimal":[72],"rounding":[73,134,145,154,159,240,246,269],"standard":[76],"LP,":[77,323],"Cohen-Addad,":[79],"Lee,":[80],"Li,":[81],"Newman":[83],"2022,":[85],"2023)":[86],"finally":[87],"bypassed":[88],"integrality":[90,316],"gap":[91,317,333],"2":[93],"for":[94,100,107,176,182,212,230,238,320],"this":[95],"LP":[96,165,191,264],"giving":[97],"1.73-approximation":[99],"problem.":[102,214],"While":[103],"introducing":[104],"new":[105],"ideas":[106],"Clustering,":[109],"their":[110],"algorithm":[111],"more":[113],"complicated":[114],"than":[115],"typical":[116,183],"approximation":[117,310,341],"algorithms":[118,135,146,241],"in":[119,227,253,306],"following":[121],"two":[122,129,273],"aspects:":[123],"(1)":[124],"It":[125,206,215],"based":[127],"on":[128],"different":[130],"relaxations":[131,211],"with":[132,342],"separate":[133],"connected":[136],"round-or-cut":[139],"procedure.":[140],"(2)":[141],"Each":[142],"has":[147],"separately":[149],"handle":[150],"seemingly":[151],"inevitable":[152],"correlated":[153,158,245],"errors,":[155],"coming":[156],"from":[157],"Sherali-Adams":[161],"other":[163,282],"strong":[164,194],"relaxations.":[166],"order":[168],"create":[170],"simple":[172,268],"unified":[174],"framework":[175,237],"similar":[179],"those":[181],"approximate":[184],"optimization":[185],"tasks,":[186],"we":[187,219,313],"propose":[188],"cluster":[190,263,322],"as":[192],"linear":[195],"program":[196],"might":[198],"tightly":[199],"capture":[200],"approximability":[202],"Clustering.":[205],"unifies":[207],"all":[208],"previous":[210],"exponential-sized,":[217],"but":[218],"show":[220,288],"it":[222],"can":[223],"be":[224,329],"(1+\u0454)-approximately":[225],"solved":[226],"polynomial":[228],"time":[229],"any":[231],"\u0454":[232],">":[233],"0,":[234],"designing":[239],"without":[242],"worrying":[243],"about":[244],"errors;":[247],"these":[248],"errors":[249],"handled":[251],"uniformly":[252],"solving":[254,283],"relaxation.":[256],"We":[257],"demonstrate":[258],"power":[260],"presenting":[266],"algorithm,":[270,304],"analyses,":[274],"one":[275],"analytically":[276],"proving":[277],"1.49-approximation":[279],"factor-revealing":[285],"SDP":[286],"1.437-approximation.":[290],"Both":[291],"proofs":[292],"introduce":[293],"principled":[294],"methods":[295],"which":[297],"analyze":[299],"performance":[301],"resulting":[305],"significantly":[308],"improved":[309,338],"guarantee.":[311],"Finally,":[312],"prove":[314],"an":[315,337],"4/3":[319],"showing":[324],"our":[325],"1.437-upper":[326],"bound":[327],"cannot":[328],"drastically":[330],"improved.":[331],"Our":[332],"instance":[334],"directly":[335],"inspires":[336],"NP-hardness":[339],"ratio":[344,351],"24/23":[345],"\u2248":[346],"1.042;":[347],"no":[348],"explicit":[349],"hardness":[350],"was":[352],"known":[353],"before.":[354]},"counts_by_year":[{"year":2026,"cited_by_count":2},{"year":2025,"cited_by_count":8}],"updated_date":"2026-07-14T23:27:15.235271","created_date":"2024-05-12T00:00:00"}
