{"id":"https://openalex.org/W2913304003","doi":"https://doi.org/10.1007/s00037-021-00211-4","title":"The hardest halfspace","display_name":"The hardest halfspace","publication_year":2021,"publication_date":"2021-08-03","ids":{"openalex":"https://openalex.org/W2913304003","doi":"https://doi.org/10.1007/s00037-021-00211-4","mag":"2913304003"},"language":"lv","primary_location":{"id":"doi:10.1007/s00037-021-00211-4","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s00037-021-00211-4","pdf_url":"https://link.springer.com/content/pdf/10.1007/s00037-021-00211-4.pdf","source":{"id":"https://openalex.org/S193995496","display_name":"Computational Complexity","issn_l":"1016-3328","issn":["1016-3328","1420-8954"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320186","host_organization_name":"Birkh\u00e4user","host_organization_lineage":["https://openalex.org/P4310320186","https://openalex.org/P4310319900"],"host_organization_lineage_names":["Birkh\u00e4user","Springer Science+Business Media"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"computational complexity","raw_type":"journal-article"},"type":"preprint","indexed_in":["arxiv","crossref","datacite"],"open_access":{"is_oa":true,"oa_status":"hybrid","oa_url":"https://link.springer.com/content/pdf/10.1007/s00037-021-00211-4.pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5079755716","display_name":"Alexander A. Sherstov","orcid":"https://orcid.org/0000-0002-2488-7852"},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Alexander A. Sherstov","raw_affiliation_strings":["Computer Science Department, UCLA, Los Angeles, CA, 90095, USA","[Computer Science Department, UCLA, Los Angeles, USA]"],"affiliations":[{"raw_affiliation_string":"Computer Science Department, UCLA, Los Angeles, CA, 90095, USA","institution_ids":[]},{"raw_affiliation_string":"[Computer Science Department, UCLA, Los Angeles, USA]","institution_ids":[]}]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5079755716"],"corresponding_institution_ids":[],"apc_list":{"value":2290,"currency":"EUR","value_usd":2890},"apc_paid":{"value":2290,"currency":"EUR","value_usd":2890},"fwci":0.0,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":{"value":0.00619497,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"30","issue":"2","first_page":"16","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":0.9993000030517578,"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":0.9993000030517578,"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":0.9987999796867371,"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/T10237","display_name":"Cryptography and Data Security","score":0.9972000122070312,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7896884679794312},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.7135228514671326},{"id":"https://openalex.org/keywords/omega","display_name":"Omega","score":0.7056178450584412},{"id":"https://openalex.org/keywords/quadratic-growth","display_name":"Quadratic growth","score":0.6466147303581238},{"id":"https://openalex.org/keywords/upper-and-lower-bounds","display_name":"Upper and lower bounds","score":0.5220608711242676},{"id":"https://openalex.org/keywords/polynomial","display_name":"Polynomial","score":0.5140970349311829},{"id":"https://openalex.org/keywords/quadratic-equation","display_name":"Quadratic equation","score":0.5138154625892639},{"id":"https://openalex.org/keywords/rank","display_name":"Rank (graph theory)","score":0.49686482548713684},{"id":"https://openalex.org/keywords/norm","display_name":"Norm (philosophy)","score":0.4542860984802246},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.34514862298965454},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.1960754096508026},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.08170592784881592},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.07276037335395813}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7896884679794312},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.7135228514671326},{"id":"https://openalex.org/C2779557605","wikidata":"https://www.wikidata.org/wiki/Q9890","display_name":"Omega","level":2,"score":0.7056178450584412},{"id":"https://openalex.org/C195956108","wikidata":"https://www.wikidata.org/wiki/Q7268362","display_name":"Quadratic growth","level":2,"score":0.6466147303581238},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.5220608711242676},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.5140970349311829},{"id":"https://openalex.org/C129844170","wikidata":"https://www.wikidata.org/wiki/Q41299","display_name":"Quadratic equation","level":2,"score":0.5138154625892639},{"id":"https://openalex.org/C164226766","wikidata":"https://www.wikidata.org/wiki/Q7293202","display_name":"Rank (graph theory)","level":2,"score":0.49686482548713684},{"id":"https://openalex.org/C191795146","wikidata":"https://www.wikidata.org/wiki/Q3878446","display_name":"Norm (philosophy)","level":2,"score":0.4542860984802246},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.34514862298965454},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.1960754096508026},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.08170592784881592},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.07276037335395813},{"id":"https://openalex.org/C199539241","wikidata":"https://www.wikidata.org/wiki/Q7748","display_name":"Law","level":1,"score":0.0},{"id":"https://openalex.org/C17744445","wikidata":"https://www.wikidata.org/wiki/Q36442","display_name":"Political science","level":0,"score":0.0},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0}],"mesh":[],"locations_count":8,"locations":[{"id":"doi:10.1007/s00037-021-00211-4","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s00037-021-00211-4","pdf_url":"https://link.springer.com/content/pdf/10.1007/s00037-021-00211-4.pdf","source":{"id":"https://openalex.org/S193995496","display_name":"Computational Complexity","issn_l":"1016-3328","issn":["1016-3328","1420-8954"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320186","host_organization_name":"Birkh\u00e4user","host_organization_lineage":["https://openalex.org/P4310320186","https://openalex.org/P4310319900"],"host_organization_lineage_names":["Birkh\u00e4user","Springer Science+Business Media"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"computational complexity","raw_type":"journal-article"},{"id":"pmh:oai:arXiv.org:1902.01765","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1902.01765","pdf_url":"https://arxiv.org/pdf/1902.01765","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":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"text"},{"id":"mag:2913304003","is_oa":true,"landing_page_url":"http://export.arxiv.org/pdf/1902.01765","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":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"arXiv (Cornell University)","raw_type":null},{"id":"pmh:oai:circle.library.ubc.ca:2429/69314","is_oa":false,"landing_page_url":"http://hdl.handle.net/2429/69314","pdf_url":null,"source":{"id":"https://openalex.org/S4377196297","display_name":"cIRcle (University of British Columbia)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I141945490","host_organization_name":"University of British Columbia","host_organization_lineage":["https://openalex.org/I141945490"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"Moving Image"},{"id":"doi:10.48550/arxiv.1902.01765","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.1902.01765","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":null,"license_id":null,"version":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"article"},{"id":"doi:10.14288/1.0377630","is_oa":true,"landing_page_url":"https://doi.org/10.14288/1.0377630","pdf_url":null,"source":{"id":"https://openalex.org/S7407053062","display_name":"Open Collections","issn_l":null,"issn":[],"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":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"article"},{"id":"doi:10.5446/59349","is_oa":true,"landing_page_url":"https://doi.org/10.5446/59349","pdf_url":null,"source":{"id":"https://openalex.org/S7407052952","display_name":"TIB KMO / FLOWWORKS GmbH","issn_l":null,"issn":[],"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":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"article"},{"id":"mag:3189036154","is_oa":false,"landing_page_url":"https://dblp.uni-trier.de/db/journals/eccc/eccc26.html#Sherstov19","pdf_url":null,"source":{"id":"https://openalex.org/S4306509107","display_name":"Electron. Colloquium Comput. Complex.","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":"journal"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":null,"raw_source_name":"Electron. Colloquium Comput. Complex.","raw_type":null}],"best_oa_location":{"id":"doi:10.1007/s00037-021-00211-4","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s00037-021-00211-4","pdf_url":"https://link.springer.com/content/pdf/10.1007/s00037-021-00211-4.pdf","source":{"id":"https://openalex.org/S193995496","display_name":"Computational Complexity","issn_l":"1016-3328","issn":["1016-3328","1420-8954"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320186","host_organization_name":"Birkh\u00e4user","host_organization_lineage":["https://openalex.org/P4310320186","https://openalex.org/P4310319900"],"host_organization_lineage_names":["Birkh\u00e4user","Springer Science+Business Media"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"computational complexity","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G101112439","display_name":null,"funder_award_id":"Fellow","funder_id":"https://openalex.org/F4320306151","funder_display_name":"Alfred P. Sloan Foundation"},{"id":"https://openalex.org/G1171700966","display_name":null,"funder_award_id":"NSF CAREER","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G2629604219","display_name":null,"funder_award_id":"CCF-1149018","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G3041994311","display_name":null,"funder_award_id":"14901","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G3658206218","display_name":"CAREER: Limits of Communication","funder_award_id":"1149018","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G6671297155","display_name":null,"funder_award_id":"CAREER","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G6894402473","display_name":null,"funder_award_id":"Fellowship","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"}],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"},{"id":"https://openalex.org/F4320306151","display_name":"Alfred P. Sloan Foundation","ror":"https://ror.org/052csg198"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2913304003.pdf","grobid_xml":"https://content.openalex.org/works/W2913304003.grobid-xml"},"referenced_works_count":82,"referenced_works":["https://openalex.org/W196871588","https://openalex.org/W1502281788","https://openalex.org/W1516726196","https://openalex.org/W1517311456","https://openalex.org/W1568582273","https://openalex.org/W1838647306","https://openalex.org/W1859250467","https://openalex.org/W1964537764","https://openalex.org/W1973728061","https://openalex.org/W1981115804","https://openalex.org/W1989966214","https://openalex.org/W1990284858","https://openalex.org/W1990948892","https://openalex.org/W2000070319","https://openalex.org/W2006822886","https://openalex.org/W2008376525","https://openalex.org/W2008671796","https://openalex.org/W2009849678","https://openalex.org/W2011320860","https://openalex.org/W2015194364","https://openalex.org/W2018199527","https://openalex.org/W2019363670","https://openalex.org/W2025070700","https://openalex.org/W2027118220","https://openalex.org/W2031374990","https://openalex.org/W2034038586","https://openalex.org/W2036908068","https://openalex.org/W2041210515","https://openalex.org/W2044859592","https://openalex.org/W2046242327","https://openalex.org/W2051576617","https://openalex.org/W2053590396","https://openalex.org/W2057648657","https://openalex.org/W2059442002","https://openalex.org/W2062223744","https://openalex.org/W2072331102","https://openalex.org/W2073505535","https://openalex.org/W2077740213","https://openalex.org/W2079401865","https://openalex.org/W2086156863","https://openalex.org/W2091623775","https://openalex.org/W2093796999","https://openalex.org/W2098730982","https://openalex.org/W2106458073","https://openalex.org/W2110503949","https://openalex.org/W2112553910","https://openalex.org/W2113364322","https://openalex.org/W2114522915","https://openalex.org/W2117822564","https://openalex.org/W2119877753","https://openalex.org/W2120210041","https://openalex.org/W2122780600","https://openalex.org/W2124500503","https://openalex.org/W2125389627","https://openalex.org/W2129275337","https://openalex.org/W2150157290","https://openalex.org/W2151303208","https://openalex.org/W2154466129","https://openalex.org/W2156569826","https://openalex.org/W2172507717","https://openalex.org/W2173499948","https://openalex.org/W2176265082","https://openalex.org/W2331905370","https://openalex.org/W2340787257","https://openalex.org/W2405936819","https://openalex.org/W2514575731","https://openalex.org/W2610386774","https://openalex.org/W2610739693","https://openalex.org/W2611287588","https://openalex.org/W2611933946","https://openalex.org/W2908369137","https://openalex.org/W2908449032","https://openalex.org/W2912052665","https://openalex.org/W2949361528","https://openalex.org/W3127719294","https://openalex.org/W4210720364","https://openalex.org/W4229606576","https://openalex.org/W4233413206","https://openalex.org/W4238893454","https://openalex.org/W4242529138","https://openalex.org/W4253594684","https://openalex.org/W6781595908"],"related_works":["https://openalex.org/W3189036154","https://openalex.org/W2402254081","https://openalex.org/W2295580398","https://openalex.org/W2953323745","https://openalex.org/W2950011701","https://openalex.org/W2949214562","https://openalex.org/W3196437393","https://openalex.org/W3099921078","https://openalex.org/W2950122418","https://openalex.org/W2093469244","https://openalex.org/W2573563314","https://openalex.org/W2949195484","https://openalex.org/W1500625764","https://openalex.org/W2553481176","https://openalex.org/W111431548","https://openalex.org/W3030539087","https://openalex.org/W2149399212","https://openalex.org/W3173219382","https://openalex.org/W2506817469","https://openalex.org/W2891222834"],"abstract_inverted_index":{"Abstract":[0],"We":[1],"study":[2],"the":[3,34,54,69,101,124,149,182],"approximation":[4,64],"of":[5,42,53,82,105,134,167,193],"halfspaces":[6],"$$h:\\{0,1\\}^n\\to\\{0,1\\}$$":[7],"<mml:math":[8,109,139,198],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">":[9,110,140,199],"<mml:mrow>":[10,14,24,113,141,200],"<mml:mi>h</mml:mi>":[11],"<mml:mo>:</mml:mo>":[12],"<mml:msup>":[13,111,205],"<mml:mo>{</mml:mo>":[15,25],"<mml:mn>0</mml:mn>":[16,26],"<mml:mo>,</mml:mo>":[17,27],"<mml:mn>1</mml:mn>":[18,28],"<mml:mo>}</mml:mo>":[19,29],"</mml:mrow>":[20,30,31,119,146,210],"<mml:mi>n</mml:mi>":[21,117,144,203,207],"</mml:msup>":[22,120,208],"<mml:mo>\u2192</mml:mo>":[23],"</mml:math>":[32,121,147,211],"in":[33],"infinity":[35],"norm":[36],"by":[37,85,170],"polynomials":[38],"and":[39,62,87,126,163,173],"rational":[40,63],"functions":[41],"any":[43],"given":[44],"degree.":[45],"Our":[46,177],"main":[47],"result":[48],"is":[49],"an":[50,91,190],"explicit":[51,191],"construction":[52],"\u201chardest\u201d":[55],"halfspace,":[56],"for":[57,74,212],"which":[58],"we":[59,93,188],"prove":[60],"polynomial":[61],"lower":[65],"bounds":[66,72],"that":[67,98],"match":[68],"trivial":[70],"upper":[71],"achievable":[73],"all":[75],"halfspaces.":[76],"This":[77],"completes":[78],"a":[79,95,132,165],"lengthy":[80],"line":[81,166],"work":[83,168],"started":[84,169],"Myhill":[86],"Kautz":[88],"(1961).":[89],"As":[90],"application,":[92],"construct":[94],"communication":[96,150,213],"problem":[97,130],"achieves":[99],"essentially":[100],"largest":[102],"possible":[103],"separation,":[104],"O(n)":[106],"versus":[107,137,154,196,216],"$$2^{-\\Omega(n)}$$":[108],"<mml:mn>2</mml:mn>":[112],"<mml:mo>-</mml:mo>":[114],"<mml:mi>\u03a9</mml:mi>":[115,142,201],"<mml:mo>(</mml:mo>":[116,143,202],"<mml:mo>)</mml:mo>":[118,145,209],",":[122],"between":[123,148],"sign-rank":[125],"discrepancy.":[127],"Equivalently,":[128],"our":[129],"exhibits":[131],"gap":[133],"log":[135,194],"n":[136,195],"$$\\Omega(n)$$":[138],"complexity":[151],"with":[152,214],"unbounded":[153,156,215,218],"weakly":[155,217],"error,":[157],"improving":[158],"quadratically":[159],"on":[160],"previous":[161],"constructions":[162],"completing":[164],"Babai,":[171],"Frankl,":[172],"Simon":[174],"(FOCS":[175],"1986).":[176],"results":[178],"further":[179],"generalize":[180],"to":[181],"k":[183],"-party":[184],"number-on-the-forehead":[185],"model,":[186],"where":[187],"obtain":[189],"separation":[192],"$$\\Omega(n/4^{n})$$":[197],"<mml:mo>/</mml:mo>":[204],"<mml:mn>4</mml:mn>":[206],"error.":[219]},"counts_by_year":[],"updated_date":"2026-04-11T08:14:18.477133","created_date":"2025-10-10T00:00:00"}
