{"@context":"https:\/\/schema.org\/","@type":"ScholarlyArticle","@id":"#article9056","name":"Constant-Factor Approximations for Asymmetric TSP on Nearly-Embeddable Graphs","abstract":"In the Asymmetric Traveling Salesperson Problem (ATSP) the goal is to find a closed walk of minimum cost in a directed graph visiting every vertex. We consider the approximability of ATSP on topologically restricted graphs. It has been shown by Oveis Gharan and Saberi [SODA, 2011] that there exists polynomial-time constant-factor approximations on planar graphs and more generally graphs of constant orientable genus. This result was extended to non-orientable genus by Erickson and Sidiropoulos [SoCG, 2014].\r\n\r\nWe show that for any class of nearly-embeddable graphs, ATSP admits a polynomial-time constant-factor approximation. More precisely, we show that for any fixed non-negative k, there exist positive alpha and beta, such that ATSP on n-vertex k-nearly-embeddable graphs admits an alpha-approximation in time O(n^beta). The class of k-nearly-embeddable graphs contains graphs with at most k apices, k vortices of width at most k, and an underlying surface of either orientable or non-orientable genus at most k. Prior to our work, even the case of graphs with a single apex was open. Our algorithm combines tools from rounding the Held-Karp LP via thin trees with dynamic programming.\r\n\r\nWe complement our upper bounds by showing that solving ATSP exactly on graphs of pathwidth k (and hence on k-nearly embeddable graphs) requires time n^{Omega(k)}, assuming the Exponential-Time Hypothesis (ETH). This is surprising in light of the fact that both TSP on undirected graphs and Minimum Cost Hamiltonian Cycle on directed graphs are FPT parameterized by treewidth.","keywords":["asymmetric TSP","approximation algorithms","nearly-embeddable graphs","Held-Karp LP","exponential time hypothesis"],"author":[{"@type":"Person","name":"Marx, D\u00e1niel","givenName":"D\u00e1niel","familyName":"Marx"},{"@type":"Person","name":"Salmasi, Ario","givenName":"Ario","familyName":"Salmasi"},{"@type":"Person","name":"Sidiropoulos, Anastasios","givenName":"Anastasios","familyName":"Sidiropoulos"}],"position":16,"pageStart":"16:1","pageEnd":"16:54","dateCreated":"2016-09-06","datePublished":"2016-09-06","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Marx, D\u00e1niel","givenName":"D\u00e1niel","familyName":"Marx"},{"@type":"Person","name":"Salmasi, Ario","givenName":"Ario","familyName":"Salmasi"},{"@type":"Person","name":"Sidiropoulos, Anastasios","givenName":"Anastasios","familyName":"Sidiropoulos"}],"copyrightYear":"2016","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.APPROX-RANDOM.2016.16","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","citation":"http:\/\/dx.doi.org\/10.1002\/net.3230120103","isPartOf":{"@type":"PublicationVolume","@id":"#volume6263","volumeNumber":60,"name":"Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX\/RANDOM 2016)","dateCreated":"2016-09-06","datePublished":"2016-09-06","editor":[{"@type":"Person","name":"Jansen, Klaus","givenName":"Klaus","familyName":"Jansen"},{"@type":"Person","name":"Mathieu, Claire","givenName":"Claire","familyName":"Mathieu"},{"@type":"Person","name":"Rolim, Jos\u00e9 D. P.","givenName":"Jos\u00e9 D. P.","familyName":"Rolim"},{"@type":"Person","name":"Umans, Chris","givenName":"Chris","familyName":"Umans"}],"isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#article9056","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":"#volume6263"}}}