{"@context":"https:\/\/schema.org\/","@type":"ScholarlyArticle","@id":"#article10236","name":"Improving TSP Tours Using Dynamic Programming over Tree Decompositions","abstract":"Given a traveling salesman problem (TSP) tour H in graph G, a k-move is an operation which removes k edges from H, and adds k edges of G so that a new tour H' is formed. The popular k-opt heuristic for TSP finds a local optimum by starting from an arbitrary tour H and then improving it by a sequence of k-moves.\r\n\r\nUntil 2016, the only known algorithm to find an improving k-move for a given tour was the naive solution in time O(n^k). At ICALP'16 de Berg, Buchin, Jansen and Woeginger showed an O(n^{floor(2\/3k)+1})-time algorithm.\r\n\r\nWe show an algorithm which runs in O(n^{(1\/4 + epsilon_k)k}) time, where lim_{k -> infinity} epsilon_k = 0. It improves over the state of the art for every k >= 5. For the most practically relevant case k=5 we provide a slightly refined algorithm running in O(n^{3.4}) time. We also show that for the k=4 case, improving over the O(n^3)-time algorithm of de Berg et al. would be a major breakthrough: an O(n^{3 - epsilon})-time algorithm for any epsilon > 0 would imply an O(n^{3 - delta})-time algorithm for the All Pairs Shortest Paths problem, for some delta>0.","keywords":["TSP","treewidth","local search","XP algorithm","hardness in P"],"author":[{"@type":"Person","name":"Cygan, Marek","givenName":"Marek","familyName":"Cygan"},{"@type":"Person","name":"Kowalik, Lukasz","givenName":"Lukasz","familyName":"Kowalik"},{"@type":"Person","name":"Socala, Arkadiusz","givenName":"Arkadiusz","familyName":"Socala"}],"position":30,"pageStart":"30:1","pageEnd":"30:14","dateCreated":"2017-09-01","datePublished":"2017-09-01","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Cygan, Marek","givenName":"Marek","familyName":"Cygan"},{"@type":"Person","name":"Kowalik, Lukasz","givenName":"Lukasz","familyName":"Kowalik"},{"@type":"Person","name":"Socala, Arkadiusz","givenName":"Arkadiusz","familyName":"Socala"}],"copyrightYear":"2017","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.ESA.2017.30","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","citation":["http:\/\/arxiv.org\/abs\/1703.05559","http:\/\/dx.doi.org\/10.4230\/LIPIcs.STACS.2010.2470","http:\/\/dx.doi.org\/10.1016\/S0377-2217(99)00284-2","http:\/\/dx.doi.org\/10.1287\/opre.21.2.498","http:\/\/dx.doi.org\/10.1137\/1033004"],"isPartOf":{"@type":"PublicationVolume","@id":"#volume6290","volumeNumber":87,"name":"25th Annual European Symposium on Algorithms (ESA 2017)","dateCreated":"2017-09-01","datePublished":"2017-09-01","editor":[{"@type":"Person","name":"Pruhs, Kirk","givenName":"Kirk","familyName":"Pruhs"},{"@type":"Person","name":"Sohler, Christian","givenName":"Christian","familyName":"Sohler"}],"isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#article10236","isPartOf":{"@type":"Periodical","@id":"#series116","name":"Leibniz International Proceedings in Informatics","issn":"1868-8969","isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#volume6290"}}}