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Parameterized Approximation Algorithms for TSP

Authors: Jianqi Zhou, Peihua Li, and Jiong Guo

Published in: LIPIcs, Volume 248, 33rd International Symposium on Algorithms and Computation (ISAAC 2022)

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.

Cite as

Jianqi Zhou, Peihua Li, and Jiong Guo. Parameterized Approximation Algorithms for TSP. In 33rd International Symposium on Algorithms and Computation (ISAAC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 248, pp. 50:1-50:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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  author =	{Zhou, Jianqi and Li, Peihua and Guo, Jiong},
  title =	{{Parameterized Approximation Algorithms for TSP}},
  booktitle =	{33rd International Symposium on Algorithms and Computation (ISAAC 2022)},
  pages =	{50:1--50:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-258-7},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{248},
  editor =	{Bae, Sang Won and Park, Heejin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2022.50},
  URN =		{urn:nbn:de:0030-drops-173358},
  doi =		{10.4230/LIPIcs.ISAAC.2022.50},
  annote =	{Keywords: FPT-approximation algorithms, the Traveling Salesman problem, the triangle inequality, fixed-parameter tractability, metric graphs}
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