{"id":"https://openalex.org/W2152299057","doi":"https://doi.org/10.1137/050627472","title":"Approximating the Radii of Point Sets","display_name":"Approximating the Radii of Point Sets","publication_year":2007,"publication_date":"2007-01-01","ids":{"openalex":"https://openalex.org/W2152299057","doi":"https://doi.org/10.1137/050627472","mag":"2152299057"},"language":"en","primary_location":{"id":"doi:10.1137/050627472","is_oa":false,"landing_page_url":"https://doi.org/10.1137/050627472","pdf_url":null,"source":{"id":"https://openalex.org/S153560523","display_name":"SIAM Journal on Computing","issn_l":"0097-5397","issn":["0097-5397","1095-7111"],"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 Computing","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/A5113712448","display_name":"Kasturi Varadarajan","orcid":null},"institutions":[{"id":"https://openalex.org/I126307644","display_name":"University of Iowa","ror":"https://ror.org/036jqmy94","country_code":"US","type":"education","lineage":["https://openalex.org/I126307644"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Kasturi Varadarajan","raw_affiliation_strings":["University of Iowa"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"University of Iowa","institution_ids":["https://openalex.org/I126307644"]}]},{"author_position":"middle","author":{"id":null,"display_name":"S. Venkatesh","orcid":null},"institutions":[{"id":"https://openalex.org/I212119943","display_name":"University of Victoria","ror":"https://ror.org/04s5mat29","country_code":"CA","type":"education","lineage":["https://openalex.org/I212119943"]}],"countries":["CA"],"is_corresponding":false,"raw_author_name":"S. Venkatesh","raw_affiliation_strings":["University of Victoria"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"University of Victoria","institution_ids":["https://openalex.org/I212119943"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5041526408","display_name":"Yinyu Ye","orcid":"https://orcid.org/0009-0001-3239-2622"},"institutions":[{"id":"https://openalex.org/I97018004","display_name":"Stanford University","ror":"https://ror.org/00f54p054","country_code":"US","type":"education","lineage":["https://openalex.org/I97018004"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Yinyu Ye","raw_affiliation_strings":["Stanford University ()"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Stanford University ()","institution_ids":["https://openalex.org/I97018004"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5100462859","display_name":"Jiawei Zhang","orcid":"https://orcid.org/0000-0003-4988-6028"},"institutions":[{"id":"https://openalex.org/I102475099","display_name":"Sup\u00e9lec","ror":"https://ror.org/00n7gwn90","country_code":"FR","type":"education","lineage":["https://openalex.org/I102475099"]},{"id":"https://openalex.org/I15196421","display_name":"University of Applied Sciences and Arts of Southern Switzerland","ror":"https://ror.org/05ep8g269","country_code":"CH","type":"education","lineage":["https://openalex.org/I15196421"]},{"id":"https://openalex.org/I4210102264","display_name":"Shandong University of Political Science and Law","ror":"https://ror.org/01b2j5886","country_code":"CN","type":"education","lineage":["https://openalex.org/I4210102264"]}],"countries":["CH","CN","FR"],"is_corresponding":false,"raw_author_name":"Jiawei Zhang","raw_affiliation_strings":["New York University<sup>**</sup>"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"New York University<sup>**</sup>","institution_ids":["https://openalex.org/I102475099","https://openalex.org/I15196421","https://openalex.org/I4210102264"]}]}],"institutions":[],"countries_distinct_count":5,"institutions_distinct_count":6,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":4.1397,"has_fulltext":false,"cited_by_count":18,"citation_normalized_percentile":{"value":0.94150315,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":90,"max":97},"biblio":{"volume":"36","issue":"6","first_page":"1764","last_page":"1776"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10996","display_name":"Computational Geometry and Mesh Generation","score":0.9998999834060669,"subfield":{"id":"https://openalex.org/subfields/1704","display_name":"Computer Graphics and Computer-Aided Design"},"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/T10996","display_name":"Computational Geometry and Mesh Generation","score":0.9998999834060669,"subfield":{"id":"https://openalex.org/subfields/1704","display_name":"Computer Graphics and Computer-Aided Design"},"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/T10374","display_name":"Advanced Graph Theory Research","score":0.996999979019165,"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":0.7138140201568604},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6476486921310425},{"id":"https://openalex.org/keywords/dimension","display_name":"Dimension (graph theory)","score":0.6250792145729065},{"id":"https://openalex.org/keywords/polynomial","display_name":"Polynomial","score":0.49607524275779724},{"id":"https://openalex.org/keywords/radius","display_name":"RADIUS","score":0.4938972592353821},{"id":"https://openalex.org/keywords/constant","display_name":"Constant (computer programming)","score":0.4890664517879486},{"id":"https://openalex.org/keywords/computational-geometry","display_name":"Computational geometry","score":0.45814037322998047},{"id":"https://openalex.org/keywords/euclidean-geometry","display_name":"Euclidean geometry","score":0.44263792037963867},{"id":"https://openalex.org/keywords/convexity","display_name":"Convexity","score":0.4118471145629883},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.3772023320198059},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.2285529375076294},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.20831197500228882},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.07955801486968994}],"concepts":[{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.7138140201568604},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6476486921310425},{"id":"https://openalex.org/C33676613","wikidata":"https://www.wikidata.org/wiki/Q13415176","display_name":"Dimension (graph theory)","level":2,"score":0.6250792145729065},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.49607524275779724},{"id":"https://openalex.org/C178635117","wikidata":"https://www.wikidata.org/wiki/Q747499","display_name":"RADIUS","level":2,"score":0.4938972592353821},{"id":"https://openalex.org/C2777027219","wikidata":"https://www.wikidata.org/wiki/Q1284190","display_name":"Constant (computer programming)","level":2,"score":0.4890664517879486},{"id":"https://openalex.org/C29123130","wikidata":"https://www.wikidata.org/wiki/Q874709","display_name":"Computational geometry","level":2,"score":0.45814037322998047},{"id":"https://openalex.org/C129782007","wikidata":"https://www.wikidata.org/wiki/Q162886","display_name":"Euclidean geometry","level":2,"score":0.44263792037963867},{"id":"https://openalex.org/C72134830","wikidata":"https://www.wikidata.org/wiki/Q5166524","display_name":"Convexity","level":2,"score":0.4118471145629883},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3772023320198059},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.2285529375076294},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.20831197500228882},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.07955801486968994},{"id":"https://openalex.org/C106159729","wikidata":"https://www.wikidata.org/wiki/Q2294553","display_name":"Financial economics","level":1,"score":0.0},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.0},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.0},{"id":"https://openalex.org/C38652104","wikidata":"https://www.wikidata.org/wiki/Q3510521","display_name":"Computer security","level":1,"score":0.0}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.1137/050627472","is_oa":false,"landing_page_url":"https://doi.org/10.1137/050627472","pdf_url":null,"source":{"id":"https://openalex.org/S153560523","display_name":"SIAM Journal on Computing","issn_l":"0097-5397","issn":["0097-5397","1095-7111"],"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 Computing","raw_type":"journal-article"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.142.6962","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.142.6962","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.stanford.edu/~yyye/radii.ps","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.154.9981","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.154.9981","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.cs.uvic.ca/~venkat/papers/radii.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.416.5174","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.416.5174","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.optimization-online.org/DB_FILE/2006/08/1447.pdf","raw_type":"text"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":36,"referenced_works":["https://openalex.org/W125811033","https://openalex.org/W572922989","https://openalex.org/W1485979075","https://openalex.org/W1552830247","https://openalex.org/W1556094370","https://openalex.org/W1691008300","https://openalex.org/W1934904397","https://openalex.org/W1970259241","https://openalex.org/W1970670479","https://openalex.org/W1975360289","https://openalex.org/W1978492002","https://openalex.org/W1979750072","https://openalex.org/W1981313592","https://openalex.org/W1984675750","https://openalex.org/W1985123706","https://openalex.org/W1999032440","https://openalex.org/W1999351098","https://openalex.org/W2019578639","https://openalex.org/W2019886197","https://openalex.org/W2021680564","https://openalex.org/W2022029673","https://openalex.org/W2025190766","https://openalex.org/W2028154682","https://openalex.org/W2035321239","https://openalex.org/W2059651397","https://openalex.org/W2069172499","https://openalex.org/W2079872371","https://openalex.org/W2086486316","https://openalex.org/W2093867071","https://openalex.org/W2143391442","https://openalex.org/W2143996311","https://openalex.org/W2148352980","https://openalex.org/W2152196442","https://openalex.org/W2171103562","https://openalex.org/W2221440952","https://openalex.org/W4241062258"],"related_works":["https://openalex.org/W2187391117","https://openalex.org/W1981809986","https://openalex.org/W1843467136","https://openalex.org/W2949367465","https://openalex.org/W4235203617","https://openalex.org/W2051545453","https://openalex.org/W2006359264","https://openalex.org/W2120764089","https://openalex.org/W4226280031","https://openalex.org/W1510883699"],"abstract_inverted_index":{"We":[0,326,348],"consider":[1],"the":[2,6,40,51,66,70,77,101,124,129,143,191,195,208,229,232,239,291,298,303,360],"problem":[3,85,94,144],"of":[4,8,30,43,58,79,164,190,197,207,223,231,241,255,279,305,346,409],"computing":[5],"outer\u2010radii":[7],"point":[9,71,80,233,306],"sets.":[10],"In":[11,266],"this":[12,267],"problem,":[13],"we":[14,120,269],"are":[15,121],"given":[16],"integers":[17],"$n,":[18],"d$,":[19,25,288],"and":[20,26,73,110,135,185,202,244,252,322,418],"k,":[21],"where":[22,63,142],"$k":[23,387],"\\le":[24],"a":[27,83,96,162,178,221,277,318,343,353,404],"set":[28,234,307],"P":[29],"n":[31],"points":[32],"in":[33,86,123,127,158,188,205],"$\\Re^d$.":[34],"The":[35,93],"goal":[36],"is":[37,50,65,82,104,132,151,177,236,311,339,371,419],"to":[38],"compute":[39],"outer":[41],"k\u2010radius":[42],"P,":[44,235],"denoted":[45],"by":[46,161,238,276,421],"${\\cal":[47],"R}_k(P)$,":[48],"which":[49,128],"minimum":[52],"over":[53],"all":[54,383],"$(d-k)$\u2010dimensional":[55],"flats":[56],"F":[57],"$\\max_{p":[59],"\\in":[60],"P}":[61],"d(p,F)$,":[62],"$d(p,F)$":[64],"Euclidean":[67],"distance":[68],"between":[69],"p":[72],"flat":[74],"F.":[75],"Computing":[76,212],"radii":[78],"sets":[81],"fundamental":[84],"computational":[87],"convexity":[88],"with":[89,317],"many":[90],"significant":[91],"applications.":[92],"admits":[95],"polynomial":[97,159,216,373],"time":[98,160,217,374],"algorithm":[99,218,310,338,375],"when":[100,173],"dimension":[102,130],"d":[103,131,389],"constant":[105,180,354],"[U.":[106],"Faigle,":[107],"W.":[108],"Kern,":[109],"M.":[111],"Streng,":[112],"Math.":[113],"Program.,":[114,247],"73":[115],"(1996),":[116],"pp.":[117,250,416],"1\u20135].":[118],"Here":[119],"interested":[122],"general":[125],"case":[126,300],"not":[133],"fixed":[134,179],"can":[136,155,273],"be":[137,156,274],"as":[138,140],"large":[139,344],"n,":[141],"becomes":[145],"NP\u2010hard":[146],"even":[147],"for":[148,168,227,282,296,342,363,382,401],"$k=1$.":[149],"It":[150],"known":[152,294,406],"that":[153,219,271,332,335,350,359,376,386],"$R_k(P)$":[154,272,378,402],"approximated":[157,275],"factor":[163,222],"$(1":[165],"+":[166],"\\varepsilon)$":[167],"any":[169,283,364],"$\\varepsilon":[170],">":[171,356],"0$":[172,357],"$d":[174],"-":[175,390],"k$":[176],"[M.":[181],"B\u0103doiu,":[182],"S.":[183,200],"Har\u2010Peled,":[184],"P.":[186,308],"Indyk,":[187],"Proceedings":[189,206],"ACM":[192,209],"Symposium":[193,210],"on":[194,211,313],"Theory":[196],"Computing,":[198],"2002;":[199],"Har\u2010Peled":[201],"K.":[203],"Varadarajan,":[204],"Geometry,":[213],"2002].":[214],"A":[215],"guarantees":[220],"$O(\\sqrt{\\log":[224,280],"n})$":[225,281],"approximation":[226,337],"$R_1(P)$,":[228],"width":[230,304],"implied":[237],"results":[240],"Nemirovski,":[242],"Roos,":[243],"Terlaky":[245],"[Math.":[246],"86":[248],"(1999),":[249],"463\u2013473]":[251],"Nesterov":[253],"[Handbook":[254],"Semidefinite":[256],"Programming":[257],"Theory,":[258],"Algorithms,":[259],"Kluwer":[260],"Academic":[261],"Publishers,":[262],"Norwell,":[263],"MA,":[264],"2000].":[265],"paper,":[268],"show":[270,349],"ratio":[278,295],"$1":[284],"\\leq":[285,287,388],"k":[286,384],"thus":[289],"matching":[290],"previously":[292,405],"best":[293],"approximating":[297],"special":[299],"$R_1":[301],"(P)$,":[302],"Our":[309,398],"based":[312],"semidefinite":[314],"programming":[315],"relaxation":[316],"new":[319],"mixed":[320],"deterministic":[321],"randomized":[323],"rounding":[324],"procedure.":[325],"also":[327],"prove":[328],"an":[329],"inapproximability":[330,399],"result":[331,400,408],"gives":[333],"evidence":[334],"our":[336],"doing":[340],"well":[341],"range":[345],"k.":[347],"there":[351,370],"exists":[352],"$\\delta":[355],"such":[358,385],"following":[361],"holds":[362],"$0":[365],"<":[366,368],"\\eps":[367],"1$:":[369],"no":[372],"approximates":[377],"within":[379],"$(\\log":[380],"n)^{\\delta}$":[381],"d^{\\varepsilon}$":[391],"unless":[392],"NP":[393],"$\\subseteq$":[394],"DTIME":[395],"$[2^{(\\log":[396],"m)^{O(1)}}]$.":[397],"extends":[403],"hardness":[407],"Brieden":[410],"[Discrete":[411],"Comput.":[412],"Geom.,":[413],"28":[414],"(2002),":[415],"201\u2013209]":[417],"proved":[420],"modifying":[422],"Brieden\u2019s":[423],"construction":[424],"using":[425],"basic":[426],"ideas":[427],"from":[428],"probabilistically":[429],"checkable":[430],"proofs":[431],"(PCP)":[432],"theory.":[433]},"counts_by_year":[{"year":2016,"cited_by_count":2},{"year":2015,"cited_by_count":1},{"year":2013,"cited_by_count":2},{"year":2012,"cited_by_count":3}],"updated_date":"2026-07-22T07:51:19.307946","created_date":"2025-10-10T00:00:00"}