{"id":"https://openalex.org/W2058258808","doi":"https://doi.org/10.1145/1806689.1806725","title":"Satisfiability allows no nontrivial sparsification unless the polynomial-time hierarchy collapses","display_name":"Satisfiability allows no nontrivial sparsification unless the polynomial-time hierarchy collapses","publication_year":2010,"publication_date":"2010-06-05","ids":{"openalex":"https://openalex.org/W2058258808","doi":"https://doi.org/10.1145/1806689.1806725","mag":"2058258808"},"language":"en","primary_location":{"id":"doi:10.1145/1806689.1806725","is_oa":false,"landing_page_url":"https://doi.org/10.1145/1806689.1806725","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Proceedings of the forty-second 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/A5052689442","display_name":"Holger Dell","orcid":"https://orcid.org/0000-0001-8955-0786"},"institutions":[{"id":"https://openalex.org/I39343248","display_name":"Humboldt-Universit\u00e4t zu Berlin","ror":"https://ror.org/01hcx6992","country_code":"DE","type":"education","lineage":["https://openalex.org/I39343248"]}],"countries":["DE"],"is_corresponding":false,"raw_author_name":"Holger Dell","raw_affiliation_strings":["Humboldt University of Berlin, Berlin, Germany","Humboldt University of Berlin, Berlin, Germany#TAB#"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Humboldt University of Berlin, Berlin, Germany","institution_ids":["https://openalex.org/I39343248"]},{"raw_affiliation_string":"Humboldt University of Berlin, Berlin, Germany#TAB#","institution_ids":["https://openalex.org/I39343248"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5111961315","display_name":"Dieter van Melkebeek","orcid":null},"institutions":[{"id":"https://openalex.org/I135310074","display_name":"University of Wisconsin\u2013Madison","ror":"https://ror.org/01y2jtd41","country_code":"US","type":"education","lineage":["https://openalex.org/I135310074"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Dieter van Melkebeek","raw_affiliation_strings":["University of Wisconsin-Madison, Madison, WI, USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"University of Wisconsin-Madison, Madison, WI, USA","institution_ids":["https://openalex.org/I135310074"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":26.9255,"has_fulltext":false,"cited_by_count":172,"citation_normalized_percentile":{"value":0.9990426,"is_in_top_1_percent":true,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":100},"biblio":{"volume":null,"issue":null,"first_page":"251","last_page":"260"},"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/T10374","display_name":"Advanced Graph Theory Research","score":0.9986000061035156,"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/T12288","display_name":"Optimization and Search Problems","score":0.9911999702453613,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/satisfiability","display_name":"Satisfiability","score":0.7275758385658264},{"id":"https://openalex.org/keywords/hierarchy","display_name":"Hierarchy","score":0.692667543888092},{"id":"https://openalex.org/keywords/time-complexity","display_name":"Time complexity","score":0.5710234045982361},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.5579765439033508},{"id":"https://openalex.org/keywords/polynomial-hierarchy","display_name":"Polynomial hierarchy","score":0.5315924286842346},{"id":"https://openalex.org/keywords/polynomial","display_name":"Polynomial","score":0.45010727643966675},{"id":"https://openalex.org/keywords/boolean-satisfiability-problem","display_name":"Boolean satisfiability problem","score":0.4178789258003235},{"id":"https://openalex.org/keywords/theoretical-computer-science","display_name":"Theoretical computer science","score":0.39960363507270813},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.39180880784988403},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.37309056520462036},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.35522231459617615}],"concepts":[{"id":"https://openalex.org/C168773769","wikidata":"https://www.wikidata.org/wiki/Q1350299","display_name":"Satisfiability","level":2,"score":0.7275758385658264},{"id":"https://openalex.org/C31170391","wikidata":"https://www.wikidata.org/wiki/Q188619","display_name":"Hierarchy","level":2,"score":0.692667543888092},{"id":"https://openalex.org/C311688","wikidata":"https://www.wikidata.org/wiki/Q2393193","display_name":"Time complexity","level":2,"score":0.5710234045982361},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.5579765439033508},{"id":"https://openalex.org/C2776013622","wikidata":"https://www.wikidata.org/wiki/Q2103021","display_name":"Polynomial hierarchy","level":3,"score":0.5315924286842346},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.45010727643966675},{"id":"https://openalex.org/C6943359","wikidata":"https://www.wikidata.org/wiki/Q875276","display_name":"Boolean satisfiability problem","level":2,"score":0.4178789258003235},{"id":"https://openalex.org/C80444323","wikidata":"https://www.wikidata.org/wiki/Q2878974","display_name":"Theoretical computer science","level":1,"score":0.39960363507270813},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.39180880784988403},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.37309056520462036},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.35522231459617615},{"id":"https://openalex.org/C34447519","wikidata":"https://www.wikidata.org/wiki/Q179522","display_name":"Market economy","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},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1145/1806689.1806725","is_oa":false,"landing_page_url":"https://doi.org/10.1145/1806689.1806725","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Proceedings of the forty-second ACM symposium on Theory of computing","raw_type":"proceedings-article"}],"best_oa_location":null,"sustainable_development_goals":[{"score":0.4099999964237213,"id":"https://metadata.un.org/sdg/16","display_name":"Peace, Justice and strong institutions"}],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":67,"referenced_works":["https://openalex.org/W36639738","https://openalex.org/W147606981","https://openalex.org/W170415682","https://openalex.org/W1480334394","https://openalex.org/W1517036688","https://openalex.org/W1517585663","https://openalex.org/W1527197079","https://openalex.org/W1539910521","https://openalex.org/W1540625694","https://openalex.org/W1544829330","https://openalex.org/W1553479936","https://openalex.org/W1559492868","https://openalex.org/W1570425181","https://openalex.org/W1596624097","https://openalex.org/W1821538950","https://openalex.org/W1909155726","https://openalex.org/W1947666077","https://openalex.org/W1965645254","https://openalex.org/W1966205523","https://openalex.org/W1970630090","https://openalex.org/W1973852671","https://openalex.org/W1974810008","https://openalex.org/W1975442866","https://openalex.org/W1977699786","https://openalex.org/W1979740015","https://openalex.org/W1986602165","https://openalex.org/W1995725694","https://openalex.org/W1998964055","https://openalex.org/W2004181701","https://openalex.org/W2006252547","https://openalex.org/W2010193278","https://openalex.org/W2012931553","https://openalex.org/W2016341899","https://openalex.org/W2033152671","https://openalex.org/W2034437384","https://openalex.org/W2036265926","https://openalex.org/W2038691311","https://openalex.org/W2051784313","https://openalex.org/W2062427087","https://openalex.org/W2066356146","https://openalex.org/W2069139349","https://openalex.org/W2079401865","https://openalex.org/W2083748805","https://openalex.org/W2087964954","https://openalex.org/W2088419249","https://openalex.org/W2107284348","https://openalex.org/W2115334306","https://openalex.org/W2118728198","https://openalex.org/W2125455032","https://openalex.org/W2144475416","https://openalex.org/W2145098470","https://openalex.org/W2159892713","https://openalex.org/W2161620758","https://openalex.org/W2405367548","https://openalex.org/W2406857838","https://openalex.org/W2483422101","https://openalex.org/W2490706771","https://openalex.org/W2595959570","https://openalex.org/W2913688336","https://openalex.org/W3008660429","https://openalex.org/W3030488835","https://openalex.org/W3138287827","https://openalex.org/W6606956742","https://openalex.org/W6632258417","https://openalex.org/W6633989784","https://openalex.org/W6644040475","https://openalex.org/W6678047303"],"related_works":["https://openalex.org/W4388962799","https://openalex.org/W3176904788","https://openalex.org/W21597398","https://openalex.org/W2091617677","https://openalex.org/W4298153058","https://openalex.org/W2408080066","https://openalex.org/W1913543287","https://openalex.org/W2268872489","https://openalex.org/W2014111643","https://openalex.org/W2127594621"],"abstract_inverted_index":{"Consider":[0],"the":[1,15,23,52,56,63,67,103,116,135,178,186,231,234,248],"following":[2],"two-player":[3],"communication":[4,61],"process":[5],"to":[6,40,46,66,107],"decide":[7,42],"a":[8,90,127,208],"language":[9],"L:":[10],"The":[11,111,196],"first":[12,64,117],"player":[13,25,65,118],"holds":[14,114,189],"entire":[16],"input":[17],"x":[18,44],"but":[19,29],"is":[20,26,39,55,97,119,122,139,212,225],"polynomially":[21],"bounded;":[22],"second":[24,68],"computationally":[27],"unbounded":[28],"does":[30],"not":[31,140],"know":[32],"any":[33,71,191],"part":[34],"of":[35,58,60,92,151,220,233,239],"x;":[36],"their":[37],"goal":[38],"cooperatively":[41],"whether":[43],"belongs":[45],"L":[47],"at":[48],"small":[49],"cost,":[50],"where":[51,228],"cost":[53,93],"measure":[54],"number":[57],"bits":[59],"from":[62],"player.":[69],"For":[70,177],"integer":[72,192],"d":[73,193],"\u2265":[74,194],"3":[75],"and":[76,121,164,256],"positive":[77],"real":[78],"\u03b5":[79,131],"we":[80],"show":[81],"that":[82,102,137,200,211],"if":[83],"satisfiability":[84],"for":[85,130,149,173,190,244],"n-variable":[86],"d-CNF":[87],"formulas":[88],"has":[89],"protocol":[91,129],"O(nd-\u03b5)":[94],"then":[95],"coNP":[96,138,224],"in":[98,141,153,159,226,247],"NP/poly,":[99,142,227],"which":[100],"implies":[101,145,199],"polynomial-time":[104],"hierarchy":[105],"collapses":[106],"its":[108],"third":[109],"level.":[110],"result":[112,144],"even":[113],"when":[115],"conondeterministic,":[120],"tight":[123,146],"as":[124],"there":[125],"exists":[126],"trivial":[128],"=":[132],"0.":[133],"Under":[134],"hypothesis":[136],"our":[143],"lower":[147],"bounds":[148],"parameters":[150],"interest":[152],"several":[154,245],"areas,":[155],"namely":[156],"sparsification,":[157],"kernelization":[158],"parameterized":[160],"complexity,":[161],"lossy":[162],"compression,":[163],"probabilistically":[165],"checkable":[166],"proofs.":[167],"By":[168],"reduction,":[169],"similar":[170],"results":[171],"hold":[172],"other":[174],"NP-complete":[175],"problems.":[176],"vertex":[179,203,251,254],"cover":[180],"problem":[181,205],"on":[182,207],"n-vertex":[183],"d-uniform":[184],"hypergraphs,":[185],"above":[187],"statement":[188],"2.":[195],"case":[197],"d=2":[198],"no":[201],"NP-hard":[202],"deletion":[204,235],"based":[206],"graph":[209],"property":[210],"inherited":[213],"by":[214],"subgraphs":[215],"can":[216],"have":[217],"kernels":[218],"consisting":[219,238],"O(k2-\u03b5)":[221],"edges":[222,241],"unless":[223],"k":[229],"denotes":[230],"size":[232],"set.":[236],"Kernels":[237],"O(k^2)":[240],"are":[242],"known":[243],"problems":[246],"class,":[249],"including":[250],"cover,":[252],"feedback":[253],"set,":[255],"bounded-degree":[257],"deletion.":[258]},"counts_by_year":[{"year":2026,"cited_by_count":2},{"year":2023,"cited_by_count":1},{"year":2022,"cited_by_count":2},{"year":2021,"cited_by_count":1},{"year":2020,"cited_by_count":5},{"year":2019,"cited_by_count":6},{"year":2018,"cited_by_count":3},{"year":2017,"cited_by_count":10},{"year":2016,"cited_by_count":13},{"year":2015,"cited_by_count":14},{"year":2014,"cited_by_count":27},{"year":2013,"cited_by_count":25},{"year":2012,"cited_by_count":37}],"updated_date":"2026-07-29T14:22:42.915294","created_date":"2025-10-10T00:00:00"}
