License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ISAAC.2022.50
URN: urn:nbn:de:0030-drops-173358
URL: https://drops.dagstuhl.de/opus/volltexte/2022/17335/
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Zhou, Jianqi ; Li, Peihua ; Guo, Jiong

Parameterized Approximation Algorithms for TSP

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LIPIcs-ISAAC-2022-50.pdf (0.7 MB)

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↦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.

BibTeX - Entry

@InProceedings{zhou_et_al:LIPIcs.ISAAC.2022.50,
  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/opus/volltexte/2022/17335},
  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}
}

Keywords: FPT-approximation algorithms, the Traveling Salesman problem, the triangle inequality, fixed-parameter tractability, metric graphs
Collection: 33rd International Symposium on Algorithms and Computation (ISAAC 2022)
Issue Date: 2022
Date of publication: 14.12.2022

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