{"id":"https://openalex.org/W1552810675","doi":"https://doi.org/10.1007/s00453-013-9801-4","title":"Worst Case and Probabilistic Analysis of the 2-Opt Algorithm for the TSP","display_name":"Worst Case and Probabilistic Analysis of the 2-Opt Algorithm for the TSP","publication_year":2013,"publication_date":"2013-06-18","ids":{"openalex":"https://openalex.org/W1552810675","doi":"https://doi.org/10.1007/s00453-013-9801-4","mag":"1552810675"},"language":"en","primary_location":{"id":"doi:10.1007/s00453-013-9801-4","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s00453-013-9801-4","pdf_url":"https://link.springer.com/content/pdf/10.1007/s00453-013-9801-4.pdf","source":{"id":"https://openalex.org/S89324355","display_name":"Algorithmica","issn_l":"0178-4617","issn":["0178-4617","1432-0541"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Algorithmica","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"hybrid","oa_url":"https://link.springer.com/content/pdf/10.1007/s00453-013-9801-4.pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5007064425","display_name":"Matthias Englert","orcid":"https://orcid.org/0000-0002-8859-7731"},"institutions":[{"id":"https://openalex.org/I39555362","display_name":"University of Warwick","ror":"https://ror.org/01a77tt86","country_code":"GB","type":"education","lineage":["https://openalex.org/I39555362"]}],"countries":["GB"],"is_corresponding":false,"raw_author_name":"Matthias Englert","raw_affiliation_strings":["DIMAP and Dept. of Computer Science, University of Warwick, Coventry, UK","DIMAP and Department of Computer Science, University of Warwick, Coventry, UK"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"DIMAP and Dept. of Computer Science, University of Warwick, Coventry, UK","institution_ids":["https://openalex.org/I39555362"]},{"raw_affiliation_string":"DIMAP and Department of Computer Science, University of Warwick, Coventry, UK","institution_ids":["https://openalex.org/I39555362"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5034122679","display_name":"Heiko R\u00f6glin","orcid":"https://orcid.org/0009-0006-8438-3986"},"institutions":[{"id":"https://openalex.org/I135140700","display_name":"University of Bonn","ror":"https://ror.org/041nas322","country_code":"DE","type":"education","lineage":["https://openalex.org/I135140700"]}],"countries":["DE"],"is_corresponding":true,"raw_author_name":"Heiko R\u00f6glin","raw_affiliation_strings":["Dept. of Computer Science, University of Bonn, Bonn, Germany","Dept. of Comput. Sci., Univ. of Bonn, Bonn, Germany"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Dept. of Computer Science, University of Bonn, Bonn, Germany","institution_ids":["https://openalex.org/I135140700"]},{"raw_affiliation_string":"Dept. of Comput. Sci., Univ. of Bonn, Bonn, Germany","institution_ids":["https://openalex.org/I135140700"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5066021048","display_name":"Berthold V\u00f6cking","orcid":null},"institutions":[{"id":"https://openalex.org/I887968799","display_name":"RWTH Aachen University","ror":"https://ror.org/04xfq0f34","country_code":"DE","type":"education","lineage":["https://openalex.org/I887968799"]}],"countries":["DE"],"is_corresponding":false,"raw_author_name":"Berthold V\u00f6cking","raw_affiliation_strings":["Dept. of Computer Science, RWTH Aachen University, Aachen, Germany","Department of Computer Science RWTH Aachen University Aachen Germany"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Dept. of Computer Science, RWTH Aachen University, Aachen, Germany","institution_ids":["https://openalex.org/I887968799"]},{"raw_affiliation_string":"Department of Computer Science RWTH Aachen University Aachen Germany","institution_ids":["https://openalex.org/I887968799"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5034122679"],"corresponding_institution_ids":["https://openalex.org/I135140700"],"apc_list":{"value":2890,"currency":"USD","value_usd":2890},"apc_paid":{"value":2890,"currency":"USD","value_usd":2890},"fwci":19.5486,"has_fulltext":true,"cited_by_count":139,"citation_normalized_percentile":{"value":0.99395964,"is_in_top_1_percent":true,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":100},"biblio":{"volume":"68","issue":"1","first_page":"190","last_page":"264"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10567","display_name":"Vehicle Routing Optimization Methods","score":0.9986000061035156,"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"}},"topics":[{"id":"https://openalex.org/T10567","display_name":"Vehicle Routing Optimization Methods","score":0.9986000061035156,"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/T11182","display_name":"Auction Theory and Applications","score":0.9977999925613403,"subfield":{"id":"https://openalex.org/subfields/1803","display_name":"Management Science and Operations Research"},"field":{"id":"https://openalex.org/fields/18","display_name":"Decision Sciences"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}},{"id":"https://openalex.org/T12288","display_name":"Optimization and Search Problems","score":0.9943000078201294,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.7143101096153259},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7011070847511292},{"id":"https://openalex.org/keywords/heuristic","display_name":"Heuristic","score":0.6706417798995972},{"id":"https://openalex.org/keywords/theory-of-computation","display_name":"Theory of computation","score":0.5942015647888184},{"id":"https://openalex.org/keywords/upper-and-lower-bounds","display_name":"Upper and lower bounds","score":0.5729433298110962},{"id":"https://openalex.org/keywords/unit-square","display_name":"Unit square","score":0.562472939491272},{"id":"https://openalex.org/keywords/probabilistic-logic","display_name":"Probabilistic logic","score":0.561730682849884},{"id":"https://openalex.org/keywords/bounded-function","display_name":"Bounded function","score":0.529338002204895},{"id":"https://openalex.org/keywords/euclidean-geometry","display_name":"Euclidean geometry","score":0.5255064964294434},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.4382815361022949},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.35103315114974976},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.23991617560386658},{"id":"https://openalex.org/keywords/statistics","display_name":"Statistics","score":0.12363746762275696},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.07826551795005798}],"concepts":[{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.7143101096153259},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7011070847511292},{"id":"https://openalex.org/C173801870","wikidata":"https://www.wikidata.org/wiki/Q201413","display_name":"Heuristic","level":2,"score":0.6706417798995972},{"id":"https://openalex.org/C24858836","wikidata":"https://www.wikidata.org/wiki/Q844718","display_name":"Theory of computation","level":2,"score":0.5942015647888184},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.5729433298110962},{"id":"https://openalex.org/C7356347","wikidata":"https://www.wikidata.org/wiki/Q1307838","display_name":"Unit square","level":2,"score":0.562472939491272},{"id":"https://openalex.org/C49937458","wikidata":"https://www.wikidata.org/wiki/Q2599292","display_name":"Probabilistic logic","level":2,"score":0.561730682849884},{"id":"https://openalex.org/C34388435","wikidata":"https://www.wikidata.org/wiki/Q2267362","display_name":"Bounded function","level":2,"score":0.529338002204895},{"id":"https://openalex.org/C129782007","wikidata":"https://www.wikidata.org/wiki/Q162886","display_name":"Euclidean geometry","level":2,"score":0.5255064964294434},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4382815361022949},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.35103315114974976},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.23991617560386658},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.12363746762275696},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.07826551795005798},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1007/s00453-013-9801-4","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s00453-013-9801-4","pdf_url":"https://link.springer.com/content/pdf/10.1007/s00453-013-9801-4.pdf","source":{"id":"https://openalex.org/S89324355","display_name":"Algorithmica","issn_l":"0178-4617","issn":["0178-4617","1432-0541"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Algorithmica","raw_type":"journal-article"}],"best_oa_location":{"id":"doi:10.1007/s00453-013-9801-4","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s00453-013-9801-4","pdf_url":"https://link.springer.com/content/pdf/10.1007/s00453-013-9801-4.pdf","source":{"id":"https://openalex.org/S89324355","display_name":"Algorithmica","issn_l":"0178-4617","issn":["0178-4617","1432-0541"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Algorithmica","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G3181115721","display_name":"Randomisation in Online Algorithms, Load Balancing and other Dynamic Problems","funder_award_id":"EP/F043333/1","funder_id":"https://openalex.org/F4320334627","funder_display_name":"Engineering and Physical Sciences Research Council"},{"id":"https://openalex.org/G7326268580","display_name":null,"funder_award_id":"WE 2842/1","funder_id":"https://openalex.org/F4320320879","funder_display_name":"Deutsche Forschungsgemeinschaft"},{"id":"https://openalex.org/G8115207947","display_name":null,"funder_award_id":"EP/F043333/1","funder_id":"https://openalex.org/F4320334627","funder_display_name":"Engineering and Physical Sciences Research Council"}],"funders":[{"id":"https://openalex.org/F4320320879","display_name":"Deutsche Forschungsgemeinschaft","ror":"https://ror.org/018mejw64"},{"id":"https://openalex.org/F4320334627","display_name":"Engineering and Physical Sciences Research Council","ror":"https://ror.org/0439y7842"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W1552810675.pdf","grobid_xml":"https://content.openalex.org/works/W1552810675.grobid-xml"},"referenced_works_count":25,"referenced_works":["https://openalex.org/W187767610","https://openalex.org/W1497676467","https://openalex.org/W1550922999","https://openalex.org/W1965978641","https://openalex.org/W1980744864","https://openalex.org/W1981110529","https://openalex.org/W1994248285","https://openalex.org/W1997988698","https://openalex.org/W2022179164","https://openalex.org/W2034053794","https://openalex.org/W2042986967","https://openalex.org/W2045173407","https://openalex.org/W2047820376","https://openalex.org/W2055569927","https://openalex.org/W2079868147","https://openalex.org/W2121956325","https://openalex.org/W2163428398","https://openalex.org/W2165142526","https://openalex.org/W2232464514","https://openalex.org/W2295428206","https://openalex.org/W2441717076","https://openalex.org/W2951571245","https://openalex.org/W2956015669","https://openalex.org/W4231916799","https://openalex.org/W4300005723"],"related_works":["https://openalex.org/W4250775169","https://openalex.org/W4309231192","https://openalex.org/W2064563243","https://openalex.org/W4360764540","https://openalex.org/W2046228374","https://openalex.org/W2963750720","https://openalex.org/W2802142240","https://openalex.org/W2157063282","https://openalex.org/W4301950763","https://openalex.org/W1992337616"],"abstract_inverted_index":{"Abstract":[0],"2-Opt":[1,87,210],"is":[2,50,128,257],"probably":[3],"the":[4,11,39,44,112,123,146,176,185,191,196,205,227,231,242,248,252,272,294],"most":[5],"basic":[6],"local":[7,180],"search":[8],"heuristic":[9,14,49],"for":[10,75,132,160,171],"TSP.":[12],"This":[13],"achieves":[15],"amazingly":[16],"good":[17],"results":[18],"on":[19,38,61,85,156,204,230,271],"\u201creal":[20],"world\u201d":[21],"Euclidean":[22,63,133],"instances":[23,64,84,101,143],"both":[24],"with":[25,218,276,298,305,312],"respect":[26,277],"to":[27,100,153,238,247,278],"running":[28,59],"time":[29,60],"and":[30,190,301],"approximation":[31,274],"ratio.":[32],"There":[33],"are":[34,106,244],"numerous":[35],"experimental":[36],"studies":[37],"performance":[40],"of":[41,81,93,126,141,179,184,188,195,208,213,234,255,269,308],"2-Opt.":[42],"However,":[43],"theoretical":[45],"knowledge":[46],"about":[47],"this":[48,71],"still":[51],"very":[52],"limited.":[53],"Not":[54],"even":[55,237],"its":[56],"worst":[57],"case":[58],"2-dimensional":[62],"was":[65,120],"known":[66],"so":[67],"far.":[68],"We":[69,135,174,199],"clarify":[70],"issue":[72],"by":[73,130,223,259],"presenting,":[74],"every":[76],"$p\\in\\mathbb{N}$":[77],",":[78,117,159],"a":[79,137,302],"family":[80],"L":[82,280],"p":[83,281],"which":[86,103,145],"can":[88,148],"take":[89],"an":[90,161,201,219,224,266],"exponential":[91],"number":[92,125,178,186,233,254],"steps.":[94],"Previous":[95],"probabilistic":[96,142,288],"analyses":[97],"were":[98],"restricted":[99],"in":[102,111,144,182],"n":[104,187],"points":[105,147,189],"placed":[107,150],"uniformly":[108],"at":[109],"random":[110],"unit":[113],"square":[114],"[0,1]":[115,157],"2":[116],"where":[118],"it":[119],"shown":[121],"that":[122,286],"expected":[124,177,206,232,253,273],"steps":[127,235,256],"bounded":[129,258],"$\\tilde{O}(n^{10})$":[131],"instances.":[134],"consider":[136],"more":[138],"advanced":[139],"model":[140,297],"be":[149],"independently":[151],"according":[152,246],"general":[154],"distributions":[155,170],"d":[158,163,316],"arbitrary":[162],"\u22652.":[164],"In":[165,262],"particular,":[166],"we":[167,264],"allow":[168],"different":[169,172],"points.":[173],"study":[175],"improvements":[181],"terms":[183],"maximal":[192],"density":[193],"\u03d5":[194,299,313],"probability":[197],"distributions.":[198],"show":[200],"upper":[202,228,267],"bound":[203,229,268],"length":[207],"any":[209],"improvement":[211],"path":[212],"$\\tilde{O}(n^{4+1/3}\\cdot\\phi^{8/3})$":[214],".":[215,240,261,317],"When":[216],"starting":[217],"initial":[220],"tour":[221],"computed":[222],"insertion":[225],"heuristic,":[226],"improves":[236],"$\\tilde{O}(n^{4+1/3-1/d}\\cdot\\phi^{8/3})$":[239],"If":[241],"distances":[243],"measured":[245],"Manhattan":[249],"metric,":[250],"then":[251],"$\\tilde{O}(n^{4-1/d}\\cdot\\phi)$":[260],"addition,":[263],"prove":[265],"$O(\\sqrt[d]{\\phi})$":[270],"factor":[275],"all":[279],"metrics.":[282],"Let":[283],"us":[284],"remark":[285],"our":[287],"analysis":[289,304],"covers":[290],"as":[291],"special":[292],"cases":[293],"uniform":[295],"input":[296],"=1":[300],"smoothed":[303],"Gaussian":[306],"perturbations":[307],"standard":[309],"deviation":[310],"\u03c3":[311,315],"\u223c1/":[314]},"counts_by_year":[{"year":2026,"cited_by_count":5},{"year":2025,"cited_by_count":15},{"year":2024,"cited_by_count":15},{"year":2023,"cited_by_count":12},{"year":2022,"cited_by_count":11},{"year":2021,"cited_by_count":7},{"year":2020,"cited_by_count":13},{"year":2019,"cited_by_count":7},{"year":2018,"cited_by_count":10},{"year":2017,"cited_by_count":2},{"year":2016,"cited_by_count":12},{"year":2015,"cited_by_count":9},{"year":2014,"cited_by_count":5},{"year":2013,"cited_by_count":4},{"year":2012,"cited_by_count":1}],"updated_date":"2026-07-31T08:31:51.225901","created_date":"2025-10-10T00:00:00"}
