{"id":"https://openalex.org/W3026895491","doi":"https://doi.org/10.3390/e22050569","title":"Uniquely Satisfiable d-Regular (k,s)-SAT Instances","display_name":"Uniquely Satisfiable d-Regular (k,s)-SAT Instances","publication_year":2020,"publication_date":"2020-05-19","ids":{"openalex":"https://openalex.org/W3026895491","doi":"https://doi.org/10.3390/e22050569","mag":"3026895491","pmid":"https://pubmed.ncbi.nlm.nih.gov/33286341"},"language":"en","primary_location":{"id":"doi:10.3390/e22050569","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e22050569","pdf_url":"https://www.mdpi.com/1099-4300/22/5/569/pdf?version=1590660603","source":{"id":"https://openalex.org/S195231649","display_name":"Entropy","issn_l":"1099-4300","issn":["1099-4300"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Entropy","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj","pubmed"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/1099-4300/22/5/569/pdf?version=1590660603","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5051095694","display_name":"Zufeng Fu","orcid":"https://orcid.org/0000-0001-6605-2370"},"institutions":[{"id":"https://openalex.org/I178232147","display_name":"Guizhou University","ror":"https://ror.org/02wmsc916","country_code":"CN","type":"education","lineage":["https://openalex.org/I178232147"]},{"id":"https://openalex.org/I4210090773","display_name":"Anshun University","ror":"https://ror.org/009jy0c86","country_code":"CN","type":"education","lineage":["https://openalex.org/I4210090773"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Zufeng Fu","raw_affiliation_strings":["College of Computer Science and Technology, Guizhou University, Guiyang 550025, China","Department of Electronics and Information Engineering, Anshun University, Anshun 561000, China"],"raw_orcid":"https://orcid.org/0000-0001-6605-2370","affiliations":[{"raw_affiliation_string":"College of Computer Science and Technology, Guizhou University, Guiyang 550025, China","institution_ids":["https://openalex.org/I178232147"]},{"raw_affiliation_string":"Department of Electronics and Information Engineering, Anshun University, Anshun 561000, China","institution_ids":["https://openalex.org/I4210090773"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5102397893","display_name":"Daoyun Xu","orcid":null},"institutions":[{"id":"https://openalex.org/I178232147","display_name":"Guizhou University","ror":"https://ror.org/02wmsc916","country_code":"CN","type":"education","lineage":["https://openalex.org/I178232147"]}],"countries":["CN"],"is_corresponding":true,"raw_author_name":"Daoyun Xu","raw_affiliation_strings":["College of Computer Science and Technology, Guizhou University, Guiyang 550025, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"College of Computer Science and Technology, Guizhou University, Guiyang 550025, China","institution_ids":["https://openalex.org/I178232147"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5102397893"],"corresponding_institution_ids":["https://openalex.org/I178232147"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2165},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2165},"fwci":0.324,"has_fulltext":true,"cited_by_count":2,"citation_normalized_percentile":{"value":0.61355294,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":94,"max":96},"biblio":{"volume":"22","issue":"5","first_page":"569","last_page":"569"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11596","display_name":"Constraint Satisfaction and Optimization","score":0.9987000226974487,"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"}},"topics":[{"id":"https://openalex.org/T11596","display_name":"Constraint Satisfaction and Optimization","score":0.9987000226974487,"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"}},{"id":"https://openalex.org/T10551","display_name":"Scheduling and Optimization Algorithms","score":0.9958000183105469,"subfield":{"id":"https://openalex.org/subfields/2209","display_name":"Industrial and Manufacturing Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11727","display_name":"Advanced Algebra and Logic","score":0.9854999780654907,"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/combinatorics","display_name":"Combinatorics","score":0.6227589249610901},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.4633157253265381},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.43934303522109985},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.3219952881336212}],"concepts":[{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.6227589249610901},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.4633157253265381},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.43934303522109985},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3219952881336212}],"mesh":[],"locations_count":5,"locations":[{"id":"doi:10.3390/e22050569","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e22050569","pdf_url":"https://www.mdpi.com/1099-4300/22/5/569/pdf?version=1590660603","source":{"id":"https://openalex.org/S195231649","display_name":"Entropy","issn_l":"1099-4300","issn":["1099-4300"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Entropy","raw_type":"journal-article"},{"id":"pmid:33286341","is_oa":false,"landing_page_url":"https://pubmed.ncbi.nlm.nih.gov/33286341","pdf_url":null,"source":{"id":"https://openalex.org/S4306525036","display_name":"PubMed","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1299303238","host_organization_name":"National Institutes of Health","host_organization_lineage":["https://openalex.org/I1299303238"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Entropy (Basel, Switzerland)","raw_type":null},{"id":"pmh:oai:doaj.org/article:73e8e0342aff4e0cbf88c24f2bceb434","is_oa":true,"landing_page_url":"https://doaj.org/article/73e8e0342aff4e0cbf88c24f2bceb434","pdf_url":null,"source":{"id":"https://openalex.org/S4306401280","display_name":"DOAJ (DOAJ: Directory of Open Access Journals)","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":"cc-by-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Entropy, Vol 22, Iss 5, p 569 (2020)","raw_type":"article"},{"id":"pmh:oai:mdpi.com:/1099-4300/22/5/569/","is_oa":true,"landing_page_url":"http://dx.doi.org/10.3390/e22050569","pdf_url":null,"source":{"id":"https://openalex.org/S4306400947","display_name":"MDPI (MDPI AG)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I4210097602","host_organization_name":"Multidisciplinary Digital Publishing Institute (Switzerland)","host_organization_lineage":["https://openalex.org/I4210097602"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Entropy","raw_type":"Text"},{"id":"pmh:oai:pubmedcentral.nih.gov:7517085","is_oa":true,"landing_page_url":"https://www.ncbi.nlm.nih.gov/pmc/articles/7517085","pdf_url":null,"source":{"id":"https://openalex.org/S2764455111","display_name":"PubMed Central","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1299303238","host_organization_name":"National Institutes of Health","host_organization_lineage":["https://openalex.org/I1299303238"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Entropy (Basel)","raw_type":"Text"}],"best_oa_location":{"id":"doi:10.3390/e22050569","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e22050569","pdf_url":"https://www.mdpi.com/1099-4300/22/5/569/pdf?version=1590660603","source":{"id":"https://openalex.org/S195231649","display_name":"Entropy","issn_l":"1099-4300","issn":["1099-4300"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Entropy","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G5248690910","display_name":"\u53d8\u5143\u6b63\u8d1f\u51fa\u73b0\u6982\u7387\u53d7\u63a7\u7684\u968f\u673a\u6b63\u5219k-SAT\u95ee\u9898\u7814\u7a76","funder_award_id":"61862051","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G5627474207","display_name":"d-\u6b63\u5219\u547d\u9898\u516c\u5f0f\u7684\u53ef\u6ee1\u8db3\u95ee\u9898\u7814\u7a76","funder_award_id":"61762019","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G7379246147","display_name":null,"funder_award_id":"61762019,61862051","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"}],"funders":[{"id":"https://openalex.org/F4320321001","display_name":"National Natural Science Foundation of China","ror":"https://ror.org/01h0zpd94"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W3026895491.pdf","grobid_xml":"https://content.openalex.org/works/W3026895491.grobid-xml"},"referenced_works_count":31,"referenced_works":["https://openalex.org/W1518705996","https://openalex.org/W1518885151","https://openalex.org/W1534168994","https://openalex.org/W1549740976","https://openalex.org/W1573449427","https://openalex.org/W1573931597","https://openalex.org/W1575350156","https://openalex.org/W1608133301","https://openalex.org/W1657855488","https://openalex.org/W1898297283","https://openalex.org/W1968198053","https://openalex.org/W1979831715","https://openalex.org/W2000171675","https://openalex.org/W2003979824","https://openalex.org/W2036265926","https://openalex.org/W2044983583","https://openalex.org/W2045502027","https://openalex.org/W2057512592","https://openalex.org/W2066022804","https://openalex.org/W2092928619","https://openalex.org/W2116648056","https://openalex.org/W2142497232","https://openalex.org/W2155218845","https://openalex.org/W2395370258","https://openalex.org/W2409909788","https://openalex.org/W2585137863","https://openalex.org/W2741122540","https://openalex.org/W2741170138","https://openalex.org/W3006801922","https://openalex.org/W3032674817","https://openalex.org/W6711567417"],"related_works":["https://openalex.org/W2899084033","https://openalex.org/W2748952813","https://openalex.org/W4391375266","https://openalex.org/W1979597421","https://openalex.org/W2007980826","https://openalex.org/W2061531152","https://openalex.org/W3002753104","https://openalex.org/W2077600819","https://openalex.org/W2142036596","https://openalex.org/W2072657027"],"abstract_inverted_index":{"Unique":[0],"k-SAT":[1,7],"is":[2,18,63,68,96,117,174,186],"the":[3,9,25,77,103,107,175,198],"promised":[4],"version":[5],"of":[6,71,76,106,114,178,201],"where":[8,73],"given":[10],"formula":[11,78,95,185],"has":[12,79],"0":[13],"or":[14],"1":[15,53,147,155,227,239,252,279],"solution":[16],"and":[17,42,84,111,148,228],"proved":[19],"to":[20,60],"be":[21],"as":[22,24],"difficult":[23],"general":[26],"k-SAT.":[27],"For":[28],"any":[29],"k":[30,38,51,81,129,135,142,151,160,170,193,222,237,248,258,267,274],"\u2265":[31,35,130,219,259],"3":[32,131],",":[33,39,54,132,136,143,152,161,171,194,223,240,249,261,268,275],"s":[34,44,90,202,218,230,250],"f":[36,133,149,168,220,272],"(":[37,43,134,141,150,159,169,192,221,229,247,266,273],"d":[40,46,137,144,153,162,172,195,224,232,269,276],")":[41,47,138,145,156,163,173,196,225,233,253,270,277],"+":[45,146,154,226,231,251,278],"/":[48,234],"2":[49,235],">":[50,236],"\u2212":[52,238],"a":[55,69,97,120,207,243],"parsimonious":[56],"reduction":[57],"from":[58],"k-CNF":[59],"d-regular":[61,93,183,210,246],"(k,s)-CNF":[62,67,94,99,184,211],"given.":[64],"Here":[65],"regular":[66,98],"subclass":[70],"CNF,":[72],"each":[74,85],"clause":[75],"exactly":[80,89],"distinct":[82],"variables,":[83],"variable":[86,116],"occurs":[87],"in":[88,101],"clauses.":[91],"A":[92],"formula,":[100],"which":[102],"absolute":[104],"value":[105,177,200],"difference":[108],"between":[109],"positive":[110],"negative":[112],"occurrences":[113],"every":[115,182],"at":[118],"most":[119],"nonnegative":[121],"integer":[122],"d.":[123],"We":[124,213],"prove":[125],"that":[126,181,204,216,264],"for":[127,217,257],"all":[128],"\u2264":[139,157,271],"u":[140,158,191,265],".":[164,280],"The":[165],"critical":[166],"function":[167],"maximal":[176],"s,":[179],"such":[180,203],"satisfiable.":[187],"In":[188],"this":[189],"study,":[190],"denotes":[197],"minimal":[199],"there":[205,241],"exists":[206,242],"uniquely":[208,244],"satisfiable":[209,245],"formula.":[212,255],"further":[214],"show":[215],"-CNF":[254],"Moreover,":[256],"7":[260],"we":[262],"have":[263]},"counts_by_year":[{"year":2022,"cited_by_count":2}],"updated_date":"2026-06-06T09:05:17.133730","created_date":"2025-10-10T00:00:00"}