{"@context":"https:\/\/schema.org\/","@type":"ScholarlyArticle","@id":"#article17385","name":"Parameterized Approximation Algorithms for TSP","abstract":"We study the Traveling Salesman problem (TSP), where given a complete undirected graph G = (V,E) with n vertices and an edge cost function c:E\u21a6R_{\u2a7e0}, the goal is to find a minimum-cost cycle visiting every vertex exactly once. It is well-known that unless P = NP, TSP cannot be approximated in polynomial time within a factor of \u03c1(n) for any computable function \u03c1, while the metric case of TSP, that the edge cost function satisfies the \u25b3-inequality, admits a polynomial-time 1.5-approximation. We investigate TSP on general graphs from the perspective of parameterized approximability. A parameterized \u03c1-approximation algorithm returns a \u03c1-approximation solution in f(k)\u22c5|I|^O(1) time, where f is a computable function and k is a parameter of the input I. We introduce two parameters, which measure the distance of a given TSP-instance from the metric case, and achieve the following two results: \r\n- A 3-approximation algorithm for TSP in O((3k\u2081)! 8^k\u2081\u22c5 n\u00b2+n\u00b3) time, where k\u2081 is the number of triangles in which the edge costs violate the \u25b3-inequality. \r\n- A 3-approximation algorithm for TSP in O(n^O(k\u2082)) time and a (6k\u2082+9)-approximation algorithm for TSP in O(k\u2082^O(k\u2082)\u22c5n\u00b3) time, where k\u2082 is the minimum number of vertices, whose removal results in a metric graph.\r\nTo our best knowledge, the above algorithms are the first non-trivial parameterized approximation algorithms for TSP on general graphs.","keywords":["FPT-approximation algorithms","the Traveling Salesman problem","the triangle inequality","fixed-parameter tractability","metric graphs"],"author":[{"@type":"Person","name":"Zhou, Jianqi","givenName":"Jianqi","familyName":"Zhou","email":"mailto:jqzhou@mail.sdu.edu.cn","affiliation":"School of Computer Science and Technology, Shandong University, Qingdao, China"},{"@type":"Person","name":"Li, Peihua","givenName":"Peihua","familyName":"Li","affiliation":"School of Computer Science and Technology, Shandong University, Qingdao, China"},{"@type":"Person","name":"Guo, Jiong","givenName":"Jiong","familyName":"Guo","email":"mailto:jguo@sdu.edu.cn","affiliation":"School of Computer Science and Technology, Shandong University, Qingdao, China"}],"position":50,"pageStart":"50:1","pageEnd":"50:16","dateCreated":"2022-12-14","datePublished":"2022-12-14","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Zhou, Jianqi","givenName":"Jianqi","familyName":"Zhou","email":"mailto:jqzhou@mail.sdu.edu.cn","affiliation":"School of Computer Science and Technology, Shandong University, Qingdao, China"},{"@type":"Person","name":"Li, Peihua","givenName":"Peihua","familyName":"Li","affiliation":"School of Computer Science and Technology, Shandong University, Qingdao, China"},{"@type":"Person","name":"Guo, Jiong","givenName":"Jiong","familyName":"Guo","email":"mailto:jguo@sdu.edu.cn","affiliation":"School of Computer Science and Technology, Shandong University, Qingdao, China"}],"copyrightYear":"2022","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.ISAAC.2022.50","funding":"The work has been supported by the National Natural Science Foundation of China (No. 61772314, 61761136017, and 62072275).","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":{"@type":"PublicationVolume","@id":"#volume6451","volumeNumber":248,"name":"33rd International Symposium on Algorithms and Computation (ISAAC 2022)","dateCreated":"2022-12-14","datePublished":"2022-12-14","editor":[{"@type":"Person","name":"Bae, Sang Won","givenName":"Sang Won","familyName":"Bae","email":"mailto:swbae@kgu.ac.kr","sameAs":"https:\/\/orcid.org\/0000-0002-8802-4247","affiliation":"Kyonggi University, Suwon, Korea"},{"@type":"Person","name":"Park, Heejin","givenName":"Heejin","familyName":"Park","email":"mailto:hjpark@hanyang.ac.kr","sameAs":"https:\/\/orcid.org\/0000-0002-8608-5994","affiliation":"Hanyang University, Seoul, Korea"}],"isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#article17385","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":"#volume6451"}}}