{"id":"https://openalex.org/W2583363911","doi":"https://doi.org/10.17169/refubium-16228","title":"The Colorful Carath\u00e9odory Problem and its Descendants","display_name":"The Colorful Carath\u00e9odory Problem and its Descendants","publication_year":2016,"publication_date":"2016-01-01","ids":{"openalex":"https://openalex.org/W2583363911","doi":"https://doi.org/10.17169/refubium-16228","mag":"2583363911"},"language":"en","primary_location":{"id":"pmh:oai:diss.fu-berlin.de:FUDISS_thesis_000000103507","is_oa":false,"landing_page_url":"http://edocs.fu-berlin.de/diss/receive/FUDISS_thesis_000000103507","pdf_url":null,"source":{"id":"https://openalex.org/S7407052913","display_name":"Universit\u00e4tsbibliothek der FU Berlin Hochschulschriftenstelle u. Dokumentenserver","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":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"","raw_type":"doc-type:doctoralThesis"},"type":"dissertation","indexed_in":["datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://refubium.fu-berlin.de/handle/fub188/12030","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5012277174","display_name":"Yannik Stein","orcid":"https://orcid.org/0000-0002-2062-7594"},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Stein, Yannik","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5012277174"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"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/T10374","display_name":"Advanced Graph Theory Research","score":0.8551999926567078,"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":0.8551999926567078,"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":0.7864000201225281,"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/mathematics","display_name":"Mathematics","score":0.43679776787757874},{"id":"https://openalex.org/keywords/mathematics-education","display_name":"Mathematics education","score":0.35720375180244446},{"id":"https://openalex.org/keywords/genealogy","display_name":"Genealogy","score":0.3539413809776306},{"id":"https://openalex.org/keywords/epistemology","display_name":"Epistemology","score":0.3527790606021881},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.33503252267837524},{"id":"https://openalex.org/keywords/philosophy","display_name":"Philosophy","score":0.304231196641922},{"id":"https://openalex.org/keywords/history","display_name":"History","score":0.26072001457214355}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.43679776787757874},{"id":"https://openalex.org/C145420912","wikidata":"https://www.wikidata.org/wiki/Q853077","display_name":"Mathematics education","level":1,"score":0.35720375180244446},{"id":"https://openalex.org/C53553401","wikidata":"https://www.wikidata.org/wiki/Q47307","display_name":"Genealogy","level":1,"score":0.3539413809776306},{"id":"https://openalex.org/C111472728","wikidata":"https://www.wikidata.org/wiki/Q9471","display_name":"Epistemology","level":1,"score":0.3527790606021881},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.33503252267837524},{"id":"https://openalex.org/C138885662","wikidata":"https://www.wikidata.org/wiki/Q5891","display_name":"Philosophy","level":0,"score":0.304231196641922},{"id":"https://openalex.org/C95457728","wikidata":"https://www.wikidata.org/wiki/Q309","display_name":"History","level":0,"score":0.26072001457214355}],"mesh":[],"locations_count":5,"locations":[{"id":"pmh:oai:diss.fu-berlin.de:FUDISS_thesis_000000103507","is_oa":false,"landing_page_url":"http://edocs.fu-berlin.de/diss/receive/FUDISS_thesis_000000103507","pdf_url":null,"source":{"id":"https://openalex.org/S7407052913","display_name":"Universit\u00e4tsbibliothek der FU Berlin Hochschulschriftenstelle u. Dokumentenserver","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":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"","raw_type":"doc-type:doctoralThesis"},{"id":"pmh:oai:refubium.fu-berlin.de:fub188/12030","is_oa":true,"landing_page_url":"https://refubium.fu-berlin.de/handle/fub188/12030","pdf_url":"https://refubium.fu-berlin.de/handle/fub188/12030","source":{"id":"https://openalex.org/S4306402444","display_name":"Refubium (Universit\u00e4tsbibliothek der Freien Universit\u00e4t Berlin)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I75951250","host_organization_name":"Freie Universit\u00e4t Berlin","host_organization_lineage":["https://openalex.org/I75951250"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"Dissertation"},{"id":"pmh:oai:aleph.bib-bvb.de:BVB01-029291323","is_oa":false,"landing_page_url":"http://www.diss.fu-berlin.de/diss/receive/FUDISS_thesis_000000103507","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"Hochschulschrift"},{"id":"doi:10.17169/refubium-16228","is_oa":true,"landing_page_url":"https://doi.org/10.17169/refubium-16228","pdf_url":null,"source":{"id":"https://openalex.org/S7407052913","display_name":"Universit\u00e4tsbibliothek der FU Berlin Hochschulschriftenstelle u. Dokumentenserver","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":null,"license_id":null,"version":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"Other"},{"id":"mag:2583363911","is_oa":false,"landing_page_url":"https://refubium.fu-berlin.de/bitstream/fub188/12030/1/PhD_thesis_Stein_Yannik.pdf","pdf_url":null,"source":null,"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":null}],"best_oa_location":{"id":"pmh:oai:refubium.fu-berlin.de:fub188/12030","is_oa":true,"landing_page_url":"https://refubium.fu-berlin.de/handle/fub188/12030","pdf_url":"https://refubium.fu-berlin.de/handle/fub188/12030","source":{"id":"https://openalex.org/S4306402444","display_name":"Refubium (Universit\u00e4tsbibliothek der Freien Universit\u00e4t Berlin)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I75951250","host_organization_name":"Freie Universit\u00e4t Berlin","host_organization_lineage":["https://openalex.org/I75951250"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"Dissertation"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2583363911.pdf"},"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W3099691229","https://openalex.org/W2299064519","https://openalex.org/W2809169079","https://openalex.org/W2132220587","https://openalex.org/W2008582047","https://openalex.org/W2504636406","https://openalex.org/W3037336089","https://openalex.org/W2061339655","https://openalex.org/W2095480278","https://openalex.org/W3104182317","https://openalex.org/W2376417508","https://openalex.org/W2952636845","https://openalex.org/W3100551532","https://openalex.org/W2301339721","https://openalex.org/W2089540941","https://openalex.org/W3125190636","https://openalex.org/W3105084242","https://openalex.org/W1990498689","https://openalex.org/W2145382810","https://openalex.org/W2285444140"],"abstract_inverted_index":{"The":[0,196,374,440],"colorful":[1,29,51,98,212,384],"Carath\u00e9odory":[2,52,99,385],"theorem":[3,7,100,214],"is":[4,126,149,203,280,378,454,470,487,504],"an":[5,249,488],"existence":[6,375],"that":[8,47,70,88,102,113,162,169,215,222,404,418,464,474,494],"implies":[9],"several":[10,243],"statements":[11],"on":[12,68,187,205],"convex":[13,123,424],"intersection":[14,174],"patterns":[15],"such":[16,87,112],"as":[17,38,200],"Tverberg's":[18],"theorem,":[19,22],"the":[20,23,28,43,50,97,114,119,128,156,173,181,211,228,238,283,383,394,420,433,437,455,468,495,502,507],"centerpoint":[21],"first":[24],"selection":[25],"lemma,":[26],"and":[27,65,139,191,278,304,353,499],"Kirchberger":[30],"theorem.":[31,53,386],"Interestingly,":[32],"these":[33],"proofs":[34],"can":[35,104,142,295,392,406,476],"be":[36,83,143,407,477],"interpreted":[37],"polynomial-time":[39,75,184,192,284],"reductions":[40,60,285],"to":[41,49,73,172,259,286,356],"ColorfulCarath\u00e9odory,":[42],"computational":[44,129],"search":[45,160,194],"problem":[46,130,328,434,496],"corresponds":[48],"We":[54,167],"exploit":[55],"this":[56],"existing":[57],"web":[58],"of":[59,131,158,175,179,198,210,230,252,271,336,376,382,427,435,444,457,506],"by":[61,320],"developing":[62],"approximation":[63,250,272,337],"algorithms":[64],"complexity":[66,182],"bounds":[67],"ColorfulCarath\u00e9odory":[69,125,148,170,199,231,253,405],"also":[71],"apply":[72],"its":[74],"descendants.":[76],"Let":[77],"C_1,":[78],"....,":[79],"C_(d+1)":[80],"\\subset":[81],"R^d":[82,449],"finite":[84],"point":[85,107,264,446],"sets":[86,347,416],"0":[89,301,350,486],"\\in":[90,94,302,351,448],"conv(C_i)":[91],"for":[92,326],"i":[93],"[d+1].":[95],"Then,":[96],"asserts":[101],"we":[103,220,241,247,256,294,331,340,354,391,402,431,472,492],"choose":[105],"one":[106,263],"from":[108,265,312,370],"each":[109,266,313,371],"set":[110,267,298,359,372,452],"C_i":[111,314,348,417],"chosen":[115],"points":[116,311,369],"C":[117,299,360,377],"contain":[118,419,465],"origin":[120,421],"in":[121,145,151,227,233,254,315,338,397,410,422,462,480],"their":[122,423],"hull.":[124],"then":[127],"finding":[132],"C.":[133],"Since":[134],"a":[135,165,201,206,224,234,297,333,358,379,445,451],"solution":[136],"always":[137,163],"exists":[138],"since":[140],"it":[141,279],"verified":[144],"polynomial":[146,399],"time,":[147],"contained":[150],"total":[152],"function":[153],"NP":[154,159],"(TFNP),":[155],"class":[157],"problems":[161],"admit":[164],"solution.":[166],"show":[168,221,403,473,493],"belongs":[171],"two":[176],"important":[177],"subclasses":[178],"TFNP:":[180],"classes":[183],"parity":[185],"argument":[186],"directed":[188],"graphs":[189],"(PPAD)":[190],"local":[193],"(PLS).":[195],"formulation":[197],"PPAD-problem":[202],"based":[204],"new":[207],"constructive":[208,244],"proof":[209],"Caratheodory":[213],"uses":[216],"Sperner's":[217],"lemma.":[218],"Moreover,":[219,401],"already":[223],"slight":[225],"change":[226],"definition":[229],"results":[232],"PLS-complete":[235],"problem.":[236],"In":[237],"second":[239],"part,":[240],"present":[242],"results.":[245],"First,":[246],"consider":[248,332,432],"version":[251],"which":[255,339],"are":[257,341],"allowed":[258],"take":[260],"more":[261],"than":[262],"C_i.":[268,373],"This":[269],"notion":[270,335],"has":[273],"not":[274],"been":[275],"studied":[276],"before":[277],"compatible":[281],"with":[282,300,349,361,460],"ColorfulCarath\u00e9odory.":[287],"For":[288],"any":[289],"fixed":[290],"eps":[291,308,484],">":[292,485],"0,":[293],"compute":[296],"conv(C)":[303],"at":[305,362],"most":[306,363],"\\lceil":[307,364],"d":[309],"\\rceil":[310,368],"d^O(eps^(-1)":[316],"log":[317],"eps^(-1))":[318],"time":[319,412,481],"repeatedly":[321],"combining":[322],"recursively":[323],"computed":[324],"approximations":[325],"lower-dimensional":[327],"instances.":[329],"Additionally,":[330],"further":[334],"given":[342,414],"only":[343,428],"k":[344,367],"<":[345],"d+1":[346],"conv(C_i),":[352],"want":[355],"find":[357],"(d+1)":[365],"/":[366],"direct":[380],"implication":[381],"Using":[387],"linear":[388],"programming":[389],"techniques,":[390],"solve":[393],"case":[395],"k=2":[396],"weakly":[398],"time.":[400],"solved":[408],"exactly":[409],"quasi-polynomial":[411],"when":[413],"poly(d)":[415],"hulls":[425],"instead":[426],"d+1.":[429],"Finally,":[430],"computing":[436],"simplicial":[438,441],"depth.":[439],"depth":[442],"sigma_P(q)":[443,475],"q":[447],"w.r.t.":[450],"P":[453,463],"number":[456],"distinct":[458],"d-simplices":[459],"vertices":[461],"q.":[466],"If":[467],"dimension":[469,503],"constant,":[471],"(1+eps)-approximated":[478],"w.h.p.":[479],"~O(n^(d/2+1)),":[482],"where":[483],"arbitrary":[489],"constant.":[490],"Furthermore,":[491],"becomes":[497],"#P-complete":[498],"W[1]-hard":[500],"if":[501],"part":[505],"input.":[508]},"counts_by_year":[],"updated_date":"2026-07-15T18:14:33.161393","created_date":"2025-10-10T00:00:00"}