{"id":"https://openalex.org/W2034534236","doi":"https://doi.org/10.1145/2213977.2214079","title":"Many sparse cuts via higher eigenvalues","display_name":"Many sparse cuts via higher eigenvalues","publication_year":2012,"publication_date":"2012-05-19","ids":{"openalex":"https://openalex.org/W2034534236","doi":"https://doi.org/10.1145/2213977.2214079","mag":"2034534236"},"language":"en","primary_location":{"id":"doi:10.1145/2213977.2214079","is_oa":false,"landing_page_url":"https://doi.org/10.1145/2213977.2214079","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Proceedings of the forty-fourth annual ACM symposium on Theory of computing","raw_type":"proceedings-article"},"type":"article","indexed_in":["arxiv","crossref"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/1111.0965","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":null,"display_name":"Anand Louis","orcid":null},"institutions":[{"id":"https://openalex.org/I130701444","display_name":"Georgia Institute of Technology","ror":"https://ror.org/01zkghx44","country_code":"US","type":"education","lineage":["https://openalex.org/I130701444"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Anand Louis","raw_affiliation_strings":["Georgia Tech, Atlanta, GA, USA"],"affiliations":[{"raw_affiliation_string":"Georgia Tech, Atlanta, GA, USA","institution_ids":["https://openalex.org/I130701444"]}]},{"author_position":"middle","author":{"id":null,"display_name":"Prasad Raghavendra","orcid":null},"institutions":[{"id":"https://openalex.org/I130701444","display_name":"Georgia Institute of Technology","ror":"https://ror.org/01zkghx44","country_code":"US","type":"education","lineage":["https://openalex.org/I130701444"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Prasad Raghavendra","raw_affiliation_strings":["Georgia Tech, Atlanta, GA, USA"],"affiliations":[{"raw_affiliation_string":"Georgia Tech, Atlanta, GA, USA","institution_ids":["https://openalex.org/I130701444"]}]},{"author_position":"middle","author":{"id":null,"display_name":"Prasad Tetali","orcid":null},"institutions":[{"id":"https://openalex.org/I130701444","display_name":"Georgia Institute of Technology","ror":"https://ror.org/01zkghx44","country_code":"US","type":"education","lineage":["https://openalex.org/I130701444"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Prasad Tetali","raw_affiliation_strings":["Georgia Tech, Atlanta, GA, USA"],"affiliations":[{"raw_affiliation_string":"Georgia Tech, Atlanta, GA, USA","institution_ids":["https://openalex.org/I130701444"]}]},{"author_position":"last","author":{"id":null,"display_name":"Santosh Vempala","orcid":null},"institutions":[{"id":"https://openalex.org/I130701444","display_name":"Georgia Institute of Technology","ror":"https://ror.org/01zkghx44","country_code":"US","type":"education","lineage":["https://openalex.org/I130701444"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Santosh Vempala","raw_affiliation_strings":["Georgia Tech, Atlanta, GA, USA"],"affiliations":[{"raw_affiliation_string":"Georgia Tech, Atlanta, GA, USA","institution_ids":["https://openalex.org/I130701444"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":4,"corresponding_author_ids":[],"corresponding_institution_ids":["https://openalex.org/I130701444"],"apc_list":null,"apc_paid":null,"fwci":5.278,"has_fulltext":false,"cited_by_count":28,"citation_normalized_percentile":{"value":0.95455545,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":98},"biblio":{"volume":null,"issue":null,"first_page":"1131","last_page":"1140"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11476","display_name":"Graph theory and applications","score":0.9995999932289124,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T11476","display_name":"Graph theory and applications","score":0.9995999932289124,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10374","display_name":"Advanced Graph Theory Research","score":0.9894999861717224,"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/T10096","display_name":"Metal-Organic Frameworks: Synthesis and Applications","score":0.9868999719619751,"subfield":{"id":"https://openalex.org/subfields/1604","display_name":"Inorganic Chemistry"},"field":{"id":"https://openalex.org/fields/16","display_name":"Chemistry"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/bounded-function","display_name":"Bounded function","score":0.6571000218391418},{"id":"https://openalex.org/keywords/eigenvalues-and-eigenvectors","display_name":"Eigenvalues and eigenvectors","score":0.6521000266075134},{"id":"https://openalex.org/keywords/disjoint-sets","display_name":"Disjoint sets","score":0.5989000201225281},{"id":"https://openalex.org/keywords/upper-and-lower-bounds","display_name":"Upper and lower bounds","score":0.5778999924659729},{"id":"https://openalex.org/keywords/laplace-operator","display_name":"Laplace operator","score":0.5076000094413757},{"id":"https://openalex.org/keywords/vertex","display_name":"Vertex (graph theory)","score":0.4878000020980835},{"id":"https://openalex.org/keywords/graph","display_name":"Graph","score":0.435699999332428},{"id":"https://openalex.org/keywords/constant","display_name":"Constant (computer programming)","score":0.43309998512268066},{"id":"https://openalex.org/keywords/laplacian-matrix","display_name":"Laplacian matrix","score":0.3993000090122223}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8792999982833862},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.7736999988555908},{"id":"https://openalex.org/C34388435","wikidata":"https://www.wikidata.org/wiki/Q2267362","display_name":"Bounded function","level":2,"score":0.6571000218391418},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.6521000266075134},{"id":"https://openalex.org/C45340560","wikidata":"https://www.wikidata.org/wiki/Q215382","display_name":"Disjoint sets","level":2,"score":0.5989000201225281},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.5778999924659729},{"id":"https://openalex.org/C165700671","wikidata":"https://www.wikidata.org/wiki/Q203484","display_name":"Laplace operator","level":2,"score":0.5076000094413757},{"id":"https://openalex.org/C80899671","wikidata":"https://www.wikidata.org/wiki/Q1304193","display_name":"Vertex (graph theory)","level":3,"score":0.4878000020980835},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.44359999895095825},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.435699999332428},{"id":"https://openalex.org/C2777027219","wikidata":"https://www.wikidata.org/wiki/Q1284190","display_name":"Constant (computer programming)","level":2,"score":0.43309998512268066},{"id":"https://openalex.org/C115178988","wikidata":"https://www.wikidata.org/wiki/Q772067","display_name":"Laplacian matrix","level":3,"score":0.3993000090122223},{"id":"https://openalex.org/C5134670","wikidata":"https://www.wikidata.org/wiki/Q1626444","display_name":"Cut","level":4,"score":0.38100001215934753},{"id":"https://openalex.org/C74003402","wikidata":"https://www.wikidata.org/wiki/Q3180727","display_name":"Spectral graph theory","level":5,"score":0.3650999963283539},{"id":"https://openalex.org/C2778945305","wikidata":"https://www.wikidata.org/wiki/Q6865502","display_name":"Minimum weight","level":2,"score":0.3587000072002411},{"id":"https://openalex.org/C57493831","wikidata":"https://www.wikidata.org/wiki/Q3134666","display_name":"Projection (relational algebra)","level":2,"score":0.3582000136375427},{"id":"https://openalex.org/C97137487","wikidata":"https://www.wikidata.org/wiki/Q729138","display_name":"Integer (computer science)","level":2,"score":0.35690000653266907},{"id":"https://openalex.org/C128669082","wikidata":"https://www.wikidata.org/wiki/Q583461","display_name":"Randomized algorithm","level":2,"score":0.3280999958515167},{"id":"https://openalex.org/C311688","wikidata":"https://www.wikidata.org/wiki/Q2393193","display_name":"Time complexity","level":2,"score":0.2897000014781952},{"id":"https://openalex.org/C63553672","wikidata":"https://www.wikidata.org/wiki/Q581168","display_name":"Binary logarithm","level":2,"score":0.28949999809265137},{"id":"https://openalex.org/C149629883","wikidata":"https://www.wikidata.org/wiki/Q660926","display_name":"Fraction (chemistry)","level":2,"score":0.28110000491142273},{"id":"https://openalex.org/C53105687","wikidata":"https://www.wikidata.org/wiki/Q173227","display_name":"Physical constant","level":2,"score":0.28049999475479126},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.25760000944137573},{"id":"https://openalex.org/C156778621","wikidata":"https://www.wikidata.org/wiki/Q1365748","display_name":"Spectrum (functional analysis)","level":2,"score":0.2551000118255615}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1145/2213977.2214079","is_oa":false,"landing_page_url":"https://doi.org/10.1145/2213977.2214079","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Proceedings of the forty-fourth annual ACM symposium on Theory of computing","raw_type":"proceedings-article"},{"id":"pmh:oai:arXiv.org:1111.0965","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1111.0965","pdf_url":"https://arxiv.org/pdf/1111.0965","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":null,"raw_type":"text"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:1111.0965","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1111.0965","pdf_url":"https://arxiv.org/pdf/1111.0965","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":null,"raw_type":"text"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":18,"referenced_works":["https://openalex.org/W1631603072","https://openalex.org/W1966461475","https://openalex.org/W1993111701","https://openalex.org/W2019404063","https://openalex.org/W2020632401","https://openalex.org/W2034331023","https://openalex.org/W2042179958","https://openalex.org/W2098350988","https://openalex.org/W2107881198","https://openalex.org/W2128550154","https://openalex.org/W2132592548","https://openalex.org/W2150148016","https://openalex.org/W2163925208","https://openalex.org/W2165732281","https://openalex.org/W2165755074","https://openalex.org/W2176446742","https://openalex.org/W3022762247","https://openalex.org/W4301425558"],"related_works":[],"abstract_inverted_index":{"Cheeger's":[0],"fundamental":[1],"inequality":[2],"states":[3],"that":[4,14,82],"any":[5,68,146],"edge-weighted":[6],"graph":[7],"has":[8],"a":[9,41,44,113,122,126,130,150,158],"vertex":[10],"subset":[11,42,151],"S":[12,152],"such":[13,81,118],"its":[15],"expansion":[16,142],"(a.k.a.":[17],"conductance)":[18],"is":[19,35,48,94,111,149,155],"bounded":[20],"as":[21],"follows:":[22],"[":[23,83],"\u03c6(S)":[24,166],"def=":[25],"(w(S,bar{S}))/(min":[26],"set(w(S),":[27],"w(bar(S))))":[28],"\u2264":[29,86,167],"\u221a(2":[30],"\u03bb2)":[31],"]":[32,91],"where":[33,92],"w":[34],"the":[36,49,54,58,63,95,100,134,139,162,175],"total":[37,163],"edge":[38],"weight":[39,154,164],"of":[40,53,57,99,121,161],"or":[43],"cut":[45],"and":[46,103,125,165],"\u03bb2":[47],"second":[50],"smallest":[51,97],"eigenvalue":[52,98],"normalized":[55,101],"Laplacian":[56,102],"graph.":[59],"Here":[60],"we":[61,132],"prove":[62],"following":[64],"natural":[65],"generalization:":[66],"for":[67,138,145],"integer":[69],"k":[70],"\u2208":[71],"[n],":[72],"there":[73,148],"exist":[74],"ck":[75],"disjoint":[76],"subsets":[77],"S1,":[78],"...,":[79],"Sck,":[80],"maxi":[84],"\u03c6(Si)":[85],"C":[87,168],"\u221a(\u03bbk":[88,169],"log":[89,170],"k)":[90],"\u03bbk":[93],"kth":[96],"c<1,C>0":[104],"are":[105,174],"suitable":[106],"absolute":[107],"constants.":[108],"Our":[109],"proof":[110],"via":[112],"polynomial-time":[114],"algorithm":[115],"to":[116,179],"find":[117],"subsets,":[119],"consisting":[120],"spectral":[123],"projection":[124],"randomized":[127],"rounding.":[128],"As":[129],"consequence,":[131],"get":[133],"same":[135],"upper":[136],"bound":[137],"small":[140],"set":[141],"problem,":[143],"namely":[144],"k,":[147],"whose":[153],"at":[156],"most":[157],"O(1/k)":[159],"fraction":[160],"k).":[171],"Both":[172],"results":[173],"best":[176],"possible":[177],"up":[178],"constant":[180],"factors.":[181]},"counts_by_year":[{"year":2025,"cited_by_count":2},{"year":2023,"cited_by_count":3},{"year":2022,"cited_by_count":2},{"year":2021,"cited_by_count":2},{"year":2020,"cited_by_count":1},{"year":2018,"cited_by_count":4},{"year":2017,"cited_by_count":1},{"year":2016,"cited_by_count":1},{"year":2015,"cited_by_count":5},{"year":2014,"cited_by_count":3},{"year":2013,"cited_by_count":2},{"year":2012,"cited_by_count":2}],"updated_date":"2026-03-20T23:20:44.827607","created_date":"2016-06-24T00:00:00"}