{"id":"https://openalex.org/W2504023209","doi":"https://doi.org/10.1137/15m1039584","title":"Partially Polynomial Kernels for Set Cover and Test Cover","display_name":"Partially Polynomial Kernels for Set Cover and Test Cover","publication_year":2016,"publication_date":"2016-01-01","ids":{"openalex":"https://openalex.org/W2504023209","doi":"https://doi.org/10.1137/15m1039584","mag":"2504023209"},"language":"en","primary_location":{"id":"doi:10.1137/15m1039584","is_oa":false,"landing_page_url":"https://doi.org/10.1137/15m1039584","pdf_url":null,"source":{"id":"https://openalex.org/S117679221","display_name":"SIAM Journal on Discrete Mathematics","issn_l":"0895-4801","issn":["0895-4801","1095-7146"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Discrete Mathematics","raw_type":"journal-article"},"type":"article","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/A5027559765","display_name":"Manu Basavaraju","orcid":"https://orcid.org/0000-0002-0814-1612"},"institutions":[{"id":"https://openalex.org/I4432739","display_name":"University of Bergen","ror":"https://ror.org/03zga2b32","country_code":"NO","type":"education","lineage":["https://openalex.org/I4432739"]}],"countries":["NO"],"is_corresponding":false,"raw_author_name":"Manu Basavaraju","raw_affiliation_strings":["University of Bergen, Norway"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"University of Bergen, Norway","institution_ids":["https://openalex.org/I4432739"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5053182876","display_name":"Mathew C. Francis","orcid":"https://orcid.org/0000-0002-0498-7856"},"institutions":[{"id":"https://openalex.org/I59611059","display_name":"Institute of Mathematical Sciences","ror":"https://ror.org/05078rg59","country_code":"IN","type":"facility","lineage":["https://openalex.org/I59611059"]},{"id":"https://openalex.org/I6498739","display_name":"Indian Statistical Institute","ror":"https://ror.org/00q2w1j53","country_code":"IN","type":"education","lineage":["https://openalex.org/I6498739"]}],"countries":["IN"],"is_corresponding":false,"raw_author_name":"Mathew C. Francis","raw_affiliation_strings":["Indian Statistical Institute (Chennai Centre), India","Institute of Mathematical Sciences, India"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Indian Statistical Institute (Chennai Centre), India","institution_ids":["https://openalex.org/I6498739"]},{"raw_affiliation_string":"Institute of Mathematical Sciences, India","institution_ids":["https://openalex.org/I59611059"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5112411811","display_name":"M. S. Ramanujan","orcid":null},"institutions":[{"id":"https://openalex.org/I2799853480","display_name":"Schloss Dagstuhl \u2013 Leibniz Center for Informatics","ror":"https://ror.org/00k4h2615","country_code":"DE","type":"facility","lineage":["https://openalex.org/I2799853480","https://openalex.org/I315704651"]}],"countries":["DE"],"is_corresponding":false,"raw_author_name":"M. S. Ramanujan","raw_affiliation_strings":["Leibniz International Proceedings in Informatics Schloss Dagstuhl -Leibniz-Zentrum f\u00fcr Informatik, Dagstuhl Publishing, Germany"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Leibniz International Proceedings in Informatics Schloss Dagstuhl -Leibniz-Zentrum f\u00fcr Informatik, Dagstuhl Publishing, Germany","institution_ids":["https://openalex.org/I2799853480"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5015806060","display_name":"Saket Saurabh","orcid":"https://orcid.org/0000-0001-7847-6402"},"institutions":[{"id":"https://openalex.org/I4432739","display_name":"University of Bergen","ror":"https://ror.org/03zga2b32","country_code":"NO","type":"education","lineage":["https://openalex.org/I4432739"]},{"id":"https://openalex.org/I59611059","display_name":"Institute of Mathematical Sciences","ror":"https://ror.org/05078rg59","country_code":"IN","type":"facility","lineage":["https://openalex.org/I59611059"]}],"countries":["IN","NO"],"is_corresponding":false,"raw_author_name":"Saket Saurabh","raw_affiliation_strings":["Institute of Mathematical Sciences, India","University of Bergen, Norway"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Institute of Mathematical Sciences, India","institution_ids":["https://openalex.org/I59611059"]},{"raw_affiliation_string":"University of Bergen, Norway","institution_ids":["https://openalex.org/I4432739"]}]}],"institutions":[],"countries_distinct_count":3,"institutions_distinct_count":4,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":12.7204,"has_fulltext":false,"cited_by_count":45,"citation_normalized_percentile":{"value":0.98732112,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":100},"biblio":{"volume":"30","issue":"3","first_page":"1401","last_page":"1423"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10374","display_name":"Advanced Graph Theory Research","score":0.9998999834060669,"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/T10374","display_name":"Advanced Graph Theory Research","score":0.9998999834060669,"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/T10720","display_name":"Complexity and Algorithms in Graphs","score":0.9994999766349792,"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/T12541","display_name":"Graph Labeling and Dimension Problems","score":0.994700014591217,"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/cover","display_name":"Cover (algebra)","score":0.8331636190414429},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.7744234800338745},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6939215660095215},{"id":"https://openalex.org/keywords/polynomial","display_name":"Polynomial","score":0.571690022945404},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.39928725361824036},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.08694383502006531}],"concepts":[{"id":"https://openalex.org/C2780428219","wikidata":"https://www.wikidata.org/wiki/Q16952335","display_name":"Cover (algebra)","level":2,"score":0.8331636190414429},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.7744234800338745},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6939215660095215},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.571690022945404},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.39928725361824036},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.08694383502006531},{"id":"https://openalex.org/C127413603","wikidata":"https://www.wikidata.org/wiki/Q11023","display_name":"Engineering","level":0,"score":0.0},{"id":"https://openalex.org/C78519656","wikidata":"https://www.wikidata.org/wiki/Q101333","display_name":"Mechanical engineering","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1137/15m1039584","is_oa":false,"landing_page_url":"https://doi.org/10.1137/15m1039584","pdf_url":null,"source":{"id":"https://openalex.org/S117679221","display_name":"SIAM Journal on Discrete Mathematics","issn_l":"0895-4801","issn":["0895-4801","1095-7146"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Discrete Mathematics","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G3207356259","display_name":null,"funder_award_id":"2007\u20132013","funder_id":"https://openalex.org/F4320320300","funder_display_name":"European Commission"},{"id":"https://openalex.org/G896728979","display_name":null,"funder_award_id":"FP/2007/2013","funder_id":"https://openalex.org/F4320320300","funder_display_name":"European Commission"}],"funders":[{"id":"https://openalex.org/F4320320300","display_name":"European Commission","ror":"https://ror.org/00k4n6c32"},{"id":"https://openalex.org/F4320320719","display_name":"Department of Science and Technology, Ministry of Science and Technology, India","ror":"https://ror.org/0101xrq71"},{"id":"https://openalex.org/F4320337380","display_name":"Division of Mathematical Sciences","ror":"https://ror.org/051fftw81"}],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":9,"referenced_works":["https://openalex.org/W1974182574","https://openalex.org/W1998964055","https://openalex.org/W2037224350","https://openalex.org/W2058293112","https://openalex.org/W2095756139","https://openalex.org/W2107284348","https://openalex.org/W2111160344","https://openalex.org/W2154399201","https://openalex.org/W3099433168"],"related_works":["https://openalex.org/W1979597421","https://openalex.org/W2007980826","https://openalex.org/W4245490552","https://openalex.org/W4225152035","https://openalex.org/W2061531152","https://openalex.org/W3002753104","https://openalex.org/W2077600819","https://openalex.org/W1587224694","https://openalex.org/W2911598644","https://openalex.org/W2042127053"],"abstract_inverted_index":{"An":[0],"instance":[1,34,153,165],"of":[2,12,38,55,69,87,100,155,168,216,245],"the":[3,7,13,31,104,110,197,242,248,251],"$(n-k)$-Set":[4,39,114,156,169,222],"Cover":[5,9,40,71,115,118,157,170,257],"or":[6],"$(n-k)$-Test":[8,70,117,159,172,256],"problems":[10],"is":[11,18,30,35,57,66,107],"form":[14],"$(\\mathcal{U},\\mathcal{S},k)$,":[15],"where":[16],"$\\mathcal{U}$":[17,56],"a":[19,36,67,93,214],"set":[20,61,94],"with":[21,26,48,79,174,261],"$n$":[22],"elements,":[23],"$\\mathcal{S}\\subseteq":[24],"2^\\mathcal{U}$":[25],"$|\\mathcal{S}|=m$,":[27],"and":[28,42,73,116,161,178,191],"$k$":[29],"parameter.":[32],"The":[33],"Yes-instance":[37,68],"if":[41,44,72,75],"only":[43,74],"there":[45,76,91],"exists":[46,77,92],"$\\mathcal{S}'\\subseteq\\mathcal{S}$":[47,78],"$|\\mathcal{S}'|\\leq":[49,80],"n-k$":[50,81],"such":[51,82,267],"that":[52,83,97,112,136,148],"every":[53],"element":[54],"contained":[58],"in":[59,62,95,109,131,196,237],"some":[60,125],"$\\mathcal{S}'$.":[63],"Similarly,":[64],"it":[65],"for":[84,221,232,255,274],"any":[85],"pair":[86],"elements":[88],"from":[89],"$\\mathcal{U}$,":[90],"$\\mathcal{S}'$":[96],"contains":[98],"one":[99],"them":[101],"but":[102],"not":[103,120],"other.":[105],"It":[106],"known":[108,195],"literature":[111],"both":[113],"do":[119,138],"admit":[121,139],"polynomial":[122,141,145,253],"kernels":[123,205],"(under":[124],"well-known":[126],"complexity":[127],"theoretic":[128],"assumptions).":[129],"However,":[130],"this":[132,233],"paper":[133],"we":[134,143,224],"show":[135],"they":[137],"\u201cpartially":[140],"kernels\u201d:":[142],"give":[144],"time":[146,238,263],"algorithms":[147],"take":[149],"as":[150],"input":[151,207],"an":[152,163,227,259,268],"$(\\mathcal{U},\\mathcal{S},k)$":[154],"(respectively,":[158,171,182],"Cover)":[160,173],"return":[162],"equivalent":[164],"$(\\tilde{\\mathcal{U}},\\tilde{\\mathcal{S}},":[166],"\\tilde{k})$":[167],"$\\tilde{k}":[175],"\\leq":[176],"k$":[177],"$|":[179],"\\tilde{\\mathcal{U}}|=":[180],"\\mathcal{O}(k^2)$":[181],"$|\\tilde{\\mathcal{U}}|=\\mathcal{O}(k^7)$).":[183],"These":[184],"results":[185,194],"allow":[186],"us":[187],"to":[188],"generalize,":[189],"improve,":[190],"unify":[192],"several":[193],"literature.":[198],"For":[199],"example,":[200],"these":[201],"immediately":[202],"imply":[203],"traditional":[204],"when":[206],"instances":[208],"satisfy":[209],"certain":[210],"\u201csparsity":[211],"properties.\u201d":[212],"Using":[213],"part":[215],"our":[217],"partial":[218],"kernelization":[219],"algorithm":[220,231,260],"Cover,":[223],"also":[225,271],"get":[226],"improved":[228],"fixed-parameter":[229],"tractable":[230],"problem":[234],"which":[235],"runs":[236],"$\\mathcal{O}(4^kk^{\\mathcal{O}(1)}(m+n)+mn)$":[239],"improving":[240],"over":[241],"previous":[243],"best":[244],"$\\mathcal{O}(8^{k+o(k)}(m+n)^{\\mathcal{O}(1)})$.":[246],"On":[247],"other":[249,275],"hand,":[250],"partially":[252],"kernel":[254],"gives":[258],"running":[262],"$\\mathcal{O}(2^{\\mathcal{O}(k^2)}(m+n)^{\\mathcal{O}(1)})$.":[264],"We":[265],"believe":[266],"approach":[269],"could":[270],"be":[272],"useful":[273],"covering":[276],"problems.":[277]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2024,"cited_by_count":2},{"year":2023,"cited_by_count":1},{"year":2021,"cited_by_count":1},{"year":2020,"cited_by_count":1},{"year":2019,"cited_by_count":5},{"year":2018,"cited_by_count":33},{"year":2016,"cited_by_count":1}],"updated_date":"2026-08-01T09:00:35.917206","created_date":"2025-10-10T00:00:00"}