{"id":"https://openalex.org/W2081684569","doi":"https://doi.org/10.1145/1374376.1374425","title":"An optimal sdp algorithm for max-cut, and equally optimal long code tests","display_name":"An optimal sdp algorithm for max-cut, and equally optimal long code tests","publication_year":2008,"publication_date":"2008-05-17","ids":{"openalex":"https://openalex.org/W2081684569","doi":"https://doi.org/10.1145/1374376.1374425","mag":"2081684569"},"language":"en","primary_location":{"id":"doi:10.1145/1374376.1374425","is_oa":false,"landing_page_url":"https://doi.org/10.1145/1374376.1374425","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Proceedings of the fortieth annual ACM symposium on Theory of computing","raw_type":"proceedings-article"},"type":"conference-paper","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/A5051167298","display_name":"Ryan O\u2019Donnell","orcid":"https://orcid.org/0009-0005-5609-5795"},"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":"Ryan O'Donnell","raw_affiliation_strings":["Carnegie Mellon University, Pittsburgh, PA, USA","Carnegie-Mellon University, Pittsburgh, Pa., USA#TAB#"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Carnegie Mellon University, Pittsburgh, PA, USA","institution_ids":["https://openalex.org/I74973139"]},{"raw_affiliation_string":"Carnegie-Mellon University, Pittsburgh, Pa., USA#TAB#","institution_ids":["https://openalex.org/I74973139"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5073986426","display_name":"Yi Wu","orcid":"https://orcid.org/0009-0006-3120-3769"},"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":"Yi Wu","raw_affiliation_strings":["Carnegie Mellon University, Pittsburgh, PA, USA","Carnegie-Mellon University, Pittsburgh, Pa., USA#TAB#"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Carnegie Mellon University, Pittsburgh, PA, USA","institution_ids":["https://openalex.org/I74973139"]},{"raw_affiliation_string":"Carnegie-Mellon University, Pittsburgh, Pa., USA#TAB#","institution_ids":["https://openalex.org/I74973139"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":[],"corresponding_institution_ids":["https://openalex.org/I74973139"],"apc_list":null,"apc_paid":null,"fwci":5.714,"has_fulltext":false,"cited_by_count":59,"citation_normalized_percentile":{"value":0.97215939,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":90,"max":99},"biblio":{"volume":"8","issue":null,"first_page":"335","last_page":"344"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10720","display_name":"Complexity and Algorithms in Graphs","score":0.9998999834060669,"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":0.9998999834060669,"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/T10374","display_name":"Advanced Graph Theory Research","score":0.9951000213623047,"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/T10142","display_name":"Formal Methods in Verification","score":0.9950000047683716,"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/rounding","display_name":"Rounding","score":0.7481990456581116},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7086937427520752},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.6405185461044312},{"id":"https://openalex.org/keywords/maximum-cut","display_name":"Maximum cut","score":0.5964028239250183},{"id":"https://openalex.org/keywords/conjecture","display_name":"Conjecture","score":0.5903830528259277},{"id":"https://openalex.org/keywords/connection","display_name":"Connection (principal bundle)","score":0.5030447840690613},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.4943389892578125},{"id":"https://openalex.org/keywords/linear-programming-relaxation","display_name":"Linear programming relaxation","score":0.4604227542877197},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.4559931457042694},{"id":"https://openalex.org/keywords/code","display_name":"Code (set theory)","score":0.4287784993648529},{"id":"https://openalex.org/keywords/graph","display_name":"Graph","score":0.4187331199645996},{"id":"https://openalex.org/keywords/function","display_name":"Function (biology)","score":0.41790491342544556},{"id":"https://openalex.org/keywords/linear-programming","display_name":"Linear programming","score":0.20295533537864685},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.1693456470966339},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.09180596470832825}],"concepts":[{"id":"https://openalex.org/C136625980","wikidata":"https://www.wikidata.org/wiki/Q663208","display_name":"Rounding","level":2,"score":0.7481990456581116},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7086937427520752},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.6405185461044312},{"id":"https://openalex.org/C165526019","wikidata":"https://www.wikidata.org/wiki/Q942557","display_name":"Maximum cut","level":3,"score":0.5964028239250183},{"id":"https://openalex.org/C2780990831","wikidata":"https://www.wikidata.org/wiki/Q319141","display_name":"Conjecture","level":2,"score":0.5903830528259277},{"id":"https://openalex.org/C13355873","wikidata":"https://www.wikidata.org/wiki/Q2920850","display_name":"Connection (principal bundle)","level":2,"score":0.5030447840690613},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.4943389892578125},{"id":"https://openalex.org/C25360446","wikidata":"https://www.wikidata.org/wiki/Q1512771","display_name":"Linear programming relaxation","level":3,"score":0.4604227542877197},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4559931457042694},{"id":"https://openalex.org/C2776760102","wikidata":"https://www.wikidata.org/wiki/Q5139990","display_name":"Code (set theory)","level":3,"score":0.4287784993648529},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.4187331199645996},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.41790491342544556},{"id":"https://openalex.org/C41045048","wikidata":"https://www.wikidata.org/wiki/Q202843","display_name":"Linear programming","level":2,"score":0.20295533537864685},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.1693456470966339},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.09180596470832825},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.0},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0},{"id":"https://openalex.org/C78458016","wikidata":"https://www.wikidata.org/wiki/Q840400","display_name":"Evolutionary biology","level":1,"score":0.0},{"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/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.1145/1374376.1374425","is_oa":false,"landing_page_url":"https://doi.org/10.1145/1374376.1374425","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Proceedings of the fortieth annual ACM symposium on Theory of computing","raw_type":"proceedings-article"},{"id":"pmh:oai:repository.cmu.edu:compsci-2152","is_oa":false,"landing_page_url":"http://repository.cmu.edu/compsci/1146","pdf_url":null,"source":{"id":"https://openalex.org/S7407050927","display_name":"KiltHub Repository","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":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Computer Science Department","raw_type":"text"},{"id":"pmh:doi:10.1184/r1/6603401","is_oa":false,"landing_page_url":null,"pdf_url":null,"source":{"id":"https://openalex.org/S4406922384","display_name":"Open MIND","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":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Journal contribution"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":48,"referenced_works":["https://openalex.org/W1511716344","https://openalex.org/W1571037334","https://openalex.org/W1604805935","https://openalex.org/W1755257097","https://openalex.org/W1868897294","https://openalex.org/W1965444148","https://openalex.org/W1970439924","https://openalex.org/W1972939134","https://openalex.org/W1975442866","https://openalex.org/W1985123706","https://openalex.org/W1985476702","https://openalex.org/W1988381072","https://openalex.org/W1990184247","https://openalex.org/W1993116423","https://openalex.org/W1999032440","https://openalex.org/W2007531540","https://openalex.org/W2014079391","https://openalex.org/W2014086530","https://openalex.org/W2016232593","https://openalex.org/W2022589731","https://openalex.org/W2026820987","https://openalex.org/W2030987833","https://openalex.org/W2036930319","https://openalex.org/W2042179958","https://openalex.org/W2043506441","https://openalex.org/W2051527336","https://openalex.org/W2058772615","https://openalex.org/W2073903404","https://openalex.org/W2085595139","https://openalex.org/W2086917808","https://openalex.org/W2087038607","https://openalex.org/W2092812429","https://openalex.org/W2093546055","https://openalex.org/W2111385661","https://openalex.org/W2132400628","https://openalex.org/W2133263321","https://openalex.org/W2143738202","https://openalex.org/W2146489710","https://openalex.org/W2149577432","https://openalex.org/W2159739530","https://openalex.org/W2167681887","https://openalex.org/W2234995292","https://openalex.org/W2401610261","https://openalex.org/W2402996186","https://openalex.org/W2914959486","https://openalex.org/W2977793662","https://openalex.org/W3023108254","https://openalex.org/W7030257799"],"related_works":["https://openalex.org/W3100114454","https://openalex.org/W2507810539","https://openalex.org/W2035016254","https://openalex.org/W2137277194","https://openalex.org/W2031277014","https://openalex.org/W4287593987","https://openalex.org/W2561769964","https://openalex.org/W3112213682","https://openalex.org/W4287552035","https://openalex.org/W1600160158"],"abstract_inverted_index":{"Let":[0],"G":[1,35],"be":[2,229],"an":[3,39,53,75,138],"undirected":[4],"graph":[5],"for":[6,34,108,143,190],"which":[7],"the":[8,20,30,43,58,98,164,175,206,242,250,254,257,268,289],"standard":[9],"Max-Cut":[10,211],"SDP":[11,54,59,76,100,168,212,261],"relaxation":[12],"achieves":[13],"at":[14,37],"least":[15],"a":[16,83,109,129,152,282],"c":[17,26,258],"fraction":[18,41],"of":[19,42,86,154,177,234,276],"total":[21,44],"edge":[22,45],"weight,":[23,46],"1/2":[24],"\u2264":[25,27],"1.":[28],"If":[29],"actual":[31],"optimal":[32,140,186,243,269],"cut":[33,244],"is":[36,52,74,126,149,219,245],"most":[38],"s":[40],"we":[47,81,103,135,184,204],"say":[48],"that":[49,147,252],"(c,":[50,72],"s)":[51,73],"gap.":[55],"We":[56,159],"define":[57],"gap":[60,101,213],"curve":[61,214],"GapSDP":[62],":":[63,71],"[1/2,1]":[64,66],"->":[65],"by":[67,96,222],"GapSDP(c)":[68,105,123],"=":[69,106],"inf{s":[70],"gap}.":[77],"In":[78,118],"this":[79,182,248],"paper":[80],"complete":[82],"long":[84],"line":[85],"work":[87],"[15,":[88],"14,":[89],"20,":[90],"36,":[91],"19,":[92],"17,":[93],"13,":[94],"28]":[95],"determining":[97],"entire":[99],"curve;":[102],"show":[104],"S(c)":[107,125,238,260],"certain":[110],"explicit":[111],"(but":[112],"complicated":[113],"to":[114,216,231],"state)":[115],"function":[116],"S.":[117],"particular,":[119],"our":[120,263],"lower":[121],"bound":[122],"-":[124,131,209,224,271],"proved":[127],"via":[128],"polynomial-time":[130,274],"RPR2'":[132],"algorithm.":[133],"Thus":[134],"have":[136,280],"given":[137,221],"efficient,":[139],"SDP-rounding":[141],"algorithm":[142,227,265,275],"Max-Cut.":[144,191],"The":[145,210],"fact":[146,251],"it":[148],"RPR2":[150,226,264],"confirms":[151],"conjecture":[153],"Feige":[155],"and":[156,162,170,200,288],"Langberg":[157],"[17].":[158],"also":[160,220],"describe":[161],"analyze":[163],"tight":[165],"connection":[166],"between":[167],"gaps":[169],"Long":[171,187],"Code":[172,188],"tests":[173,189],"(and":[174],"constructions":[176],"[25,":[178],"3,":[179],"4]).":[180],"Using":[181],"connection,":[183],"give":[185],"Combining":[192],"these":[193],"with":[194,249],"results":[195],"implicit":[196],"in":[197,239,253],"[27,":[198],"29]":[199],"ideas":[201],"from":[202],"[19],":[203],"derive":[205],"following":[207],"conclusions:":[208],"subject":[215],"triangle":[217],"inequalities":[218],"S(c).":[223],"No":[225],"can":[228,279],"guaranteed":[230],"find":[232],"cuts":[233],"value":[235],"larger":[236],"than":[237],"graphs":[240,255],"where":[241],"c.":[246],"(Contrast":[247],"exhibiting":[256],"vs.":[259],"gap,":[262],"actually":[266],"finds":[267],"cut.)":[270],"Further,":[272],"no":[273],"any":[277],"kind":[278],"such":[281],"guarantee,":[283],"assuming":[284],"P":[285],"\u2260":[286],"NP":[287],"Unique":[290],"Games":[291],"Conjecture.":[292]},"counts_by_year":[{"year":2023,"cited_by_count":1},{"year":2022,"cited_by_count":4},{"year":2021,"cited_by_count":3},{"year":2020,"cited_by_count":2},{"year":2019,"cited_by_count":2},{"year":2018,"cited_by_count":3},{"year":2016,"cited_by_count":3},{"year":2015,"cited_by_count":1},{"year":2014,"cited_by_count":4},{"year":2013,"cited_by_count":5},{"year":2012,"cited_by_count":9}],"updated_date":"2026-07-29T14:22:42.915294","created_date":"2025-10-10T00:00:00"}
