2026-07-22T14:05:58Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:17335 2024-03-06T10:59:29Z ddc:004 open_access

Parameterized Approximation Algorithms for TSP Zhou, Jianqi Li, Peihua Guo, Jiong FPT-approximation algorithms the Traveling Salesman problem the triangle inequality fixed-parameter tractability metric graphs 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↦R_{⩾0}, 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 ρ(n) for any computable function ρ, while the metric case of TSP, that the edge cost function satisfies the △-inequality, admits a polynomial-time 1.5-approximation. We investigate TSP on general graphs from the perspective of parameterized approximability. A parameterized ρ-approximation algorithm returns a ρ-approximation solution in f(k)⋅|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: - A 3-approximation algorithm for TSP in O((3k₁)! 8^k₁⋅ n²+n³) time, where k₁ is the number of triangles in which the edge costs violate the △-inequality. - A 3-approximation algorithm for TSP in O(n^O(k₂)) time and a (6k₂+9)-approximation algorithm for TSP in O(k₂^O(k₂)⋅n³) time, where k₂ is the minimum number of vertices, whose removal results in a metric graph. To our best knowledge, the above algorithms are the first non-trivial parameterized approximation algorithms for TSP on general graphs. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Jianqi Zhou and Peihua Li and Jiong Guo 2022 Is Part Of LIPIcs, Volume 248, 33rd International Symposium on Algorithms and Computation (ISAAC 2022) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.ISAAC.2022.50 urn:nbn:de:0030-drops-173358 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2022.50 eng https://creativecommons.org/licenses/by/4.0/legalcode