{"id":"https://openalex.org/W7155373820","doi":"https://doi.org/10.48550/arxiv.2604.19851","title":"Is Four Enough? Automated Reasoning Approaches and Dual Bounds for Condorcet Dimensions of Elections","display_name":"Is Four Enough? Automated Reasoning Approaches and Dual Bounds for Condorcet Dimensions of Elections","publication_year":2026,"publication_date":"2026-04-21","ids":{"openalex":"https://openalex.org/W7155373820","doi":"https://doi.org/10.48550/arxiv.2604.19851"},"language":null,"primary_location":{"id":"doi:10.48550/arxiv.2604.19851","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2604.19851","pdf_url":null,"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":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"article"},"type":"preprint","indexed_in":["datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://doi.org/10.48550/arxiv.2604.19851","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5025876479","display_name":"Itai Zilberstein","orcid":"https://orcid.org/0009-0008-7175-5319"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Zilberstein, Itai","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5134436607","display_name":"Ratip Emin Berker","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Berker, Ratip Emin","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5134369227","display_name":"George Li","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Li, George","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5134405322","display_name":"Ruben Martins","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Martins, Ruben","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":4,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":false,"cited_by_count":0,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10991","display_name":"Game Theory and Voting Systems","score":0.9890999794006348,"subfield":{"id":"https://openalex.org/subfields/2002","display_name":"Economics and Econometrics"},"field":{"id":"https://openalex.org/fields/20","display_name":"Economics, Econometrics and Finance"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}},"topics":[{"id":"https://openalex.org/T10991","display_name":"Game Theory and Voting Systems","score":0.9890999794006348,"subfield":{"id":"https://openalex.org/subfields/2002","display_name":"Economics and Econometrics"},"field":{"id":"https://openalex.org/fields/20","display_name":"Economics, Econometrics and Finance"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}},{"id":"https://openalex.org/T10720","display_name":"Complexity and Algorithms in Graphs","score":0.0031999999191612005,"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/T11596","display_name":"Constraint Satisfaction and Optimization","score":0.00139999995008111,"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/condorcet-method","display_name":"Condorcet method","score":0.9455999732017517},{"id":"https://openalex.org/keywords/impossibility","display_name":"Impossibility","score":0.6305999755859375},{"id":"https://openalex.org/keywords/conjecture","display_name":"Conjecture","score":0.5249999761581421},{"id":"https://openalex.org/keywords/linear-programming","display_name":"Linear programming","score":0.5170000195503235},{"id":"https://openalex.org/keywords/rank","display_name":"Rank (graph theory)","score":0.5038999915122986},{"id":"https://openalex.org/keywords/upper-and-lower-bounds","display_name":"Upper and lower bounds","score":0.4936000108718872},{"id":"https://openalex.org/keywords/set","display_name":"Set (abstract data type)","score":0.48190000653266907},{"id":"https://openalex.org/keywords/social-choice-theory","display_name":"Social choice theory","score":0.45010000467300415}],"concepts":[{"id":"https://openalex.org/C140940377","wikidata":"https://www.wikidata.org/wiki/Q868720","display_name":"Condorcet method","level":4,"score":0.9455999732017517},{"id":"https://openalex.org/C2776261394","wikidata":"https://www.wikidata.org/wiki/Q315562","display_name":"Impossibility","level":2,"score":0.6305999755859375},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5252000093460083},{"id":"https://openalex.org/C2780990831","wikidata":"https://www.wikidata.org/wiki/Q319141","display_name":"Conjecture","level":2,"score":0.5249999761581421},{"id":"https://openalex.org/C41045048","wikidata":"https://www.wikidata.org/wiki/Q202843","display_name":"Linear programming","level":2,"score":0.5170000195503235},{"id":"https://openalex.org/C164226766","wikidata":"https://www.wikidata.org/wiki/Q7293202","display_name":"Rank (graph theory)","level":2,"score":0.5038999915122986},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.4936000108718872},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.48190000653266907},{"id":"https://openalex.org/C189352744","wikidata":"https://www.wikidata.org/wiki/Q1971461","display_name":"Social choice theory","level":2,"score":0.45010000467300415},{"id":"https://openalex.org/C144237770","wikidata":"https://www.wikidata.org/wiki/Q747534","display_name":"Mathematical economics","level":1,"score":0.4489000141620636},{"id":"https://openalex.org/C33676613","wikidata":"https://www.wikidata.org/wiki/Q13415176","display_name":"Dimension (graph theory)","level":2,"score":0.4352000057697296},{"id":"https://openalex.org/C2780980858","wikidata":"https://www.wikidata.org/wiki/Q110022","display_name":"Dual (grammatical number)","level":2,"score":0.420199990272522},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.3864000141620636},{"id":"https://openalex.org/C100479058","wikidata":"https://www.wikidata.org/wiki/Q526530","display_name":"Approval voting","level":5,"score":0.33500000834465027},{"id":"https://openalex.org/C136979486","wikidata":"https://www.wikidata.org/wiki/Q773483","display_name":"Existential quantification","level":2,"score":0.3154999911785126},{"id":"https://openalex.org/C189430467","wikidata":"https://www.wikidata.org/wiki/Q7293293","display_name":"Ranking (information retrieval)","level":2,"score":0.3075000047683716},{"id":"https://openalex.org/C2776029896","wikidata":"https://www.wikidata.org/wiki/Q3935810","display_name":"Relaxation (psychology)","level":2,"score":0.2994000017642975},{"id":"https://openalex.org/C311688","wikidata":"https://www.wikidata.org/wiki/Q2393193","display_name":"Time complexity","level":2,"score":0.2937000095844269},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.29280000925064087},{"id":"https://openalex.org/C17231256","wikidata":"https://www.wikidata.org/wiki/Q5156540","display_name":"Completeness (order theory)","level":2,"score":0.2838999927043915},{"id":"https://openalex.org/C2776036281","wikidata":"https://www.wikidata.org/wiki/Q48769818","display_name":"Constraint (computer-aided design)","level":2,"score":0.27900001406669617},{"id":"https://openalex.org/C80444323","wikidata":"https://www.wikidata.org/wiki/Q2878974","display_name":"Theoretical computer science","level":1,"score":0.27070000767707825},{"id":"https://openalex.org/C520049643","wikidata":"https://www.wikidata.org/wiki/Q189760","display_name":"Voting","level":3,"score":0.2653999924659729},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.25949999690055847},{"id":"https://openalex.org/C25360446","wikidata":"https://www.wikidata.org/wiki/Q1512771","display_name":"Linear programming relaxation","level":3,"score":0.2547999918460846},{"id":"https://openalex.org/C108710211","wikidata":"https://www.wikidata.org/wiki/Q11538","display_name":"Mathematical proof","level":2,"score":0.2506999969482422},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.2500999867916107}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.48550/arxiv.2604.19851","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2604.19851","pdf_url":null,"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":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"article"}],"best_oa_location":{"id":"doi:10.48550/arxiv.2604.19851","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2604.19851","pdf_url":null,"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":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"article"},"sustainable_development_goals":[{"id":"https://metadata.un.org/sdg/16","display_name":"Peace, Justice and strong institutions","score":0.5221241116523743}],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":0,"referenced_works":[],"related_works":[],"abstract_inverted_index":{"In":[0],"an":[1,77,123],"election":[2,203,256],"where":[3],"$n$":[4],"voters":[5,28],"rank":[6],"$m$":[7],"candidates,":[8],"a":[9,14,25,45,66,133,152,184,188,205,227,234,272,282],"Condorcet":[10,41,118,252],"winning":[11,42,119,235],"set":[12,67,236],"is":[13],"committee":[15,31,206],"of":[16,27,68,98,154,178,237,254],"$k$":[17],"candidates":[18,99],"such":[19,156],"that":[20,36,65,142,233,250,291],"for":[21,56,72,95,117,140,267,275],"any":[22,202,255],"outside":[23],"candidate,":[24],"majority":[26],"prefer":[29],"some":[30,37],"member.":[32],"Condorcet's":[33],"paradox":[34],"shows":[35],"elections":[38,141,277],"admit":[39],"no":[40],"sets":[43],"with":[44],"single":[46],"candidate":[47],"(i.e.,":[48],"$k=1$),":[49],"and":[50,89,114,161,168,225,281],"the":[51,59,82,96,108,111,166,176,179,221,251],"same":[52],"can":[53],"be":[54,258],"shown":[55],"$k=2$.":[57],"On":[58],"other":[60],"hand,":[61],"recent":[62],"work":[63],"proves":[64],"size":[69,209,238],"$k=5$":[70],"exists":[71],"every":[73],"election.":[74],"This":[75],"leaves":[76],"important":[78],"theoretical":[79],"gap":[80,109],"between":[81,110],"best":[83],"known":[84,262],"lower":[85],"bound":[86,91],"$(k\\geq":[87],"3)$":[88],"upper":[90,190,263],"$(k":[92],"\\leq":[93],"5)$":[94],"number":[97,153],"needed":[100],"to":[101,106,127,138,147,164,200],"guarantee":[102],"existence.":[103],"We":[104,121,131,150,270],"aim":[105],"close":[107],"existence":[112],"guarantees":[113],"impossibility":[115],"results":[116,215,245],"sets.":[120],"explore":[122],"automated":[124,243],"reasoning":[125,244],"approach":[126],"tighten":[128],"these":[129],"bounds.":[130,149],"design":[132],"mixed-integer":[134],"linear":[135,180,223],"program":[136,224],"(MILP)":[137],"search":[139,167,194],"would":[143],"serve":[144],"as":[145,157,183],"counter-examples":[146],"conjectured":[148],"employ":[151],"optimizations,":[155],"symmetry":[158],"breaking,":[159],"subsampling,":[160],"constraint":[162],"generation,":[163],"enhance":[165],"model":[169],"effectively":[170],"infinite":[171],"electorates.":[172],"Furthermore,":[173],"we":[174,198,219],"analyze":[175],"dual":[177,222],"programming":[181],"relaxation":[182],"path":[185,286],"towards":[186],"obtaining":[187],"new":[189],"bound.":[191],"Despite":[192],"extensive":[193],"on":[195],"moderate-sized":[196],"elections,":[197],"fail":[199],"find":[201],"requiring":[204],"larger":[207],"than":[208,260],"3.":[210],"Motivated":[211],"by":[212],"our":[213],"experimental":[214],"in":[216,278],"this":[217],"direction,":[218],"simplify":[220],"formulate":[226],"conjecture":[228],"which,":[229],"if":[230],"true,":[231],"implies":[232],"4":[239],"always":[240],"exists.":[241],"Our":[242],"provide":[246],"strong":[247],"empirical":[248],"evidence":[249],"dimension":[253],"may":[257],"smaller":[259,292],"currently":[261],"bounds,":[264],"at":[265],"least":[266],"small":[268],"instances.":[269],"offer":[271],"general-purpose":[273],"framework":[274],"searching":[276],"ranked":[279],"voting":[280],"new,":[283],"concrete":[284],"analytical":[285],"via":[287],"duality":[288],"toward":[289],"proving":[290],"committees":[293],"suffice.":[294]},"counts_by_year":[],"updated_date":"2026-06-11T09:08:48.828518","created_date":"2026-04-24T00:00:00"}
