{"id":"https://openalex.org/W7225003055","doi":"https://doi.org/10.5281/zenodo.23825724","title":"Improved Blocky Partition, Cover, and Equality-Rank Bounds for Set Disjointness via Kronecker Rebalancing","display_name":"Improved Blocky Partition, Cover, and Equality-Rank Bounds for Set Disjointness via Kronecker Rebalancing","publication_year":2026,"publication_date":"2026-10-31","ids":{"openalex":"https://openalex.org/W7225003055","doi":"https://doi.org/10.5281/zenodo.23825724"},"language":"en","primary_location":{"id":"doi:10.5281/zenodo.23825724","doi":"https://doi.org/10.5281/zenodo.23825724","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.23825724","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"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":"Preprint"},"volume":null,"type":"preprint","indexed_in":["datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://doi.org/10.5281/zenodo.23825724","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5160046767","display_name":"Kevin Grey","orcid":"https://orcid.org/0009-0007-8755-6501","observed_orcids":["https://orcid.org/0009-0007-8755-6501"]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Grey, Kevin","raw_affiliation_strings":["Independent Researcher"],"raw_orcid":"https://orcid.org/0009-0007-8755-6501","affiliations":[{"raw_affiliation_string":"Independent Researcher","institution_ids":[]}]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5160046767"],"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/T10720","display_name":"Complexity and Algorithms in Graphs","score":0.8931000232696533,"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":0.8931000232696533,"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":0.06849999725818634,"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/T10948","display_name":"Advanced Combinatorial Mathematics","score":0.013199999928474426,"subfield":{"id":"https://openalex.org/subfields/2607","display_name":"Discrete Mathematics and Combinatorics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/set-disjointness","display_name":"set disjointness","score":0.7990000247955322},{"id":"https://openalex.org/keywords/kronecker-product","display_name":"Kronecker product","score":0.753000020980835},{"id":"https://openalex.org/keywords/orthogonal-vectors","display_name":"orthogonal vectors","score":0.4399999976158142},{"id":"https://openalex.org/keywords/communication-complexity","display_name":"communication complexity","score":0.42899999022483826},{"id":"https://openalex.org/keywords/lower-bounds","display_name":"lower bounds","score":0.3050000071525574}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8043000102043152},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.6926000118255615},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.5483999848365784},{"id":"https://openalex.org/C34388435","wikidata":"https://www.wikidata.org/wiki/Q2267362","display_name":"Bounded function","level":2,"score":0.5098999738693237},{"id":"https://openalex.org/C2780428219","wikidata":"https://www.wikidata.org/wiki/Q16952335","display_name":"Cover (algebra)","level":2,"score":0.5029000043869019},{"id":"https://openalex.org/C42812","wikidata":"https://www.wikidata.org/wiki/Q1082910","display_name":"Partition (number theory)","level":2,"score":0.4982999861240387},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.49140000343322754},{"id":"https://openalex.org/C39482219","wikidata":"https://www.wikidata.org/wiki/Q192826","display_name":"Kronecker delta","level":2,"score":0.4142000079154968},{"id":"https://openalex.org/C46030957","wikidata":"https://www.wikidata.org/wiki/Q1238125","display_name":"Kronecker product","level":3,"score":0.3889999985694885},{"id":"https://openalex.org/C85817219","wikidata":"https://www.wikidata.org/wiki/Q884772","display_name":"Block matrix","level":3,"score":0.36629998683929443},{"id":"https://openalex.org/C2781302577","wikidata":"https://www.wikidata.org/wiki/Q209","display_name":"Rectangle","level":2,"score":0.35690000653266907},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.34940001368522644},{"id":"https://openalex.org/C48372109","wikidata":"https://www.wikidata.org/wiki/Q3913","display_name":"Binary number","level":2,"score":0.3409000039100647},{"id":"https://openalex.org/C164226766","wikidata":"https://www.wikidata.org/wiki/Q7293202","display_name":"Rank (graph theory)","level":2,"score":0.31540000438690186},{"id":"https://openalex.org/C9652623","wikidata":"https://www.wikidata.org/wiki/Q190109","display_name":"Field (mathematics)","level":2,"score":0.2989000082015991},{"id":"https://openalex.org/C115973184","wikidata":"https://www.wikidata.org/wiki/Q245457","display_name":"Circulant matrix","level":2,"score":0.29159998893737793},{"id":"https://openalex.org/C80899671","wikidata":"https://www.wikidata.org/wiki/Q1304193","display_name":"Vertex (graph theory)","level":3,"score":0.28139999508857727},{"id":"https://openalex.org/C17349429","wikidata":"https://www.wikidata.org/wiki/Q1049914","display_name":"Matrix multiplication","level":3,"score":0.27709999680519104},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.27379998564720154},{"id":"https://openalex.org/C45340560","wikidata":"https://www.wikidata.org/wiki/Q215382","display_name":"Disjoint sets","level":2,"score":0.27070000767707825},{"id":"https://openalex.org/C187834632","wikidata":"https://www.wikidata.org/wiki/Q188804","display_name":"Factorization","level":2,"score":0.26910001039505005},{"id":"https://openalex.org/C163561899","wikidata":"https://www.wikidata.org/wiki/Q1994977","display_name":"Logical matrix","level":3,"score":0.2689000070095062},{"id":"https://openalex.org/C136625980","wikidata":"https://www.wikidata.org/wiki/Q663208","display_name":"Rounding","level":2,"score":0.25949999690055847},{"id":"https://openalex.org/C184898388","wikidata":"https://www.wikidata.org/wiki/Q1435712","display_name":"Pairwise comparison","level":2,"score":0.25600001215934753},{"id":"https://openalex.org/C177148314","wikidata":"https://www.wikidata.org/wiki/Q170084","display_name":"Generalization","level":2,"score":0.2549000084400177}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.5281/zenodo.23825724","doi":"https://doi.org/10.5281/zenodo.23825724","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.23825724","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"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":"Preprint"},{"id":"doi:10.5281/zenodo.23825725","doi":"https://doi.org/10.5281/zenodo.23825725","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.23825725","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"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":"Preprint"}],"best_oa_location":{"id":"doi:10.5281/zenodo.23825724","doi":"https://doi.org/10.5281/zenodo.23825724","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.23825724","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"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":"Preprint"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W7224993173","https://openalex.org/W7171874503","https://openalex.org/W7221611410","https://openalex.org/W6945521677","https://openalex.org/W7099230457","https://openalex.org/W3130137236","https://openalex.org/W7171702855","https://openalex.org/W4414944818","https://openalex.org/W2968870016","https://openalex.org/W1966840469"],"abstract_inverted_index":{"A":[0],"binary":[1],"matrix":[2],"$B\\in\\{0,1\\}^{m\\times":[3],"n}$":[4],"is":[5],"blocky":[6,29,42,170],"(an":[7],"equality":[8,35,48],"matrix)":[9],"if":[10],"its":[11],"$1$-entries":[12],"form":[13],"pairwise":[14],"row-":[15],"and":[16,33,41,46,60,65,70,74,85,91,100,136,169,180,193],"column-disjoint":[17],"rectangles.":[18],"For":[19],"Set":[20],"Disjointness":[21],"$D_d=D_1^{\\otimes":[22],"d}\\in\\{0,1\\}^{2^d\\times":[23],"2^d}$,":[24],"Williams":[25],"(FOCS":[26],"2024)":[27],"bounded":[28],"cover":[30,105,138],"number":[31,44],"$\\mathrm{bc}(D_d)$":[32,101],"weak":[34],"rank":[36,49],"$\\mathrm{weq}_{\\mathbb{R}}(D_d)$":[37],"by":[38,51],"$O(6^{d/6})\\approx":[39],"O(1.3480^d)$":[40],"partition":[43],"$\\mathrm{bp}(D_d)$":[45,99],"exact":[47,104,166],"$\\mathrm{br}_R(D_d)$":[50,190],"$O(5^{d/5})\\approx":[52],"O(1.3797^d)$.":[53],"While":[54],"multi-bucket":[55],"separator":[56],"circuits":[57,77],"give":[58],"$\\tilde{O}(n\\cdot(2/\\sqrt{3})^d)$-time":[59],"-space":[61],"Orthogonal":[62],"Vectors":[63],"(Nederlof":[64],"W\\k{e}grzycki,":[66],"STOC":[67],"2021;":[68],"Alman":[69],"Li,":[71,86],"FOCS":[72],"2025)":[73],"unrestricted":[75],"depth-$2$":[76],"solve":[78],"$\\#\\mathrm{OV}$":[79],"in":[80,109],"$o(n\\cdot":[81],"1.25^d)$":[82],"time":[83],"(Alman":[84],"2025;":[87],"Ye,":[88],"2026),":[89],"$O(5^{d/5})$":[90],"$O(6^{d/6})$":[92],"remained":[93],"the":[94,103,144],"best":[95],"published":[96],"bounds":[97,189],"on":[98,147],"(with":[102],"base":[106],"$2/\\sqrt{3}$":[107],"established":[108],"a":[110,159],"companion":[111],"paper).":[112],"First,":[113],"via":[114],"nonnegative":[115],"LP":[116,163],"dual":[117],"vertex":[118],"potentials":[119],"we":[120,157],"prove":[121],"$\\mathrm{fbc}(D_d)=\\mathrm{bp}^*(D_d)$":[122],"for":[123],"$1\\le":[124],"d\\le":[125],"5$":[126],"with":[127],"values":[128],"$(3/2,7/4,23/10,35/13,205/62)$":[129],"(showing":[130],"$\\mathrm{bc}(D_5),\\mathrm{bp}(D_5)\\in\\{4,5\\}$,":[131],"alongside":[132],"$\\mathrm{bp}(D_4)=4$);":[133],"Kronecker":[134,161],"submultiplicativity":[135],"greedy":[137],"rounding":[139],"yield":[140],"$\\mathrm{bc}(D_d)=O(d\\cdot(205/62)^{d/5})=O(d\\cdot":[141],"1.27021^d)$":[142],"(and":[143],"same":[145],"bound":[146],"$\\mathrm{weq}_{\\mathbb{F}}(D_d)$":[148],"over":[149,165,195],"every":[150,196],"infinite":[151],"or":[152],"$>3^d$-sized":[153],"field":[154,197],"$\\mathbb{F}$).":[155],"Second,":[156],"introduce":[158],"palette":[160],"rebalancing":[162],"$\\tau_{\\mathrm{bp}}(M)$":[164],"rectangle":[167],"partitions":[168],"packings,":[171],"proving":[172],"$\\tau_{\\mathrm{bp}}(D_1,D_2,D_3)=(3/2,11/6,23/10)$,":[173],"$\\mathrm{bp}(A\\otimes":[174],"D_3)\\le":[175],"276\\lceil\\mathrm{bp}(A)/120\\rceil$":[176],"(refined":[177],"to":[178],"$92\\lceil\\mathrm{bp}(A)/40\\rceil$),":[179],"$\\mathrm{bp}(D_d)<3\\cdot(23/10)^{d/3}<3\\cdot":[181],"1.320007^d$":[182],"($<11\\cdot(23/10)^{d/3}$":[183],"from":[184],"$120\\to":[185],"276$":[186],"alone),":[187],"which":[188],"($R\\ne":[191],"0$)":[192],"$\\mathrm{spr}_{\\mathbb{F}}(D_d),\\mathrm{weq}_{\\mathbb{F}}(D_d)$":[194],"$\\mathbb{F}$.":[198]},"counts_by_year":[],"updated_date":"2026-11-02T09:45:03.842371","created_date":"2026-11-01T00:00:00"}