License: Creative Commons Attribution 3.0 Unported license (CC-BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ICALP.2018.9
URN: urn:nbn:de:0030-drops-90133
URL: https://drops.dagstuhl.de/opus/volltexte/2018/9013/
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Adamaszek, Anna ; Mnich, Matthias ; Paluch, Katarzyna

New Approximation Algorithms for (1,2)-TSP

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Abstract

We give faster and simpler approximation algorithms for the (1,2)-TSP problem, a well-studied variant of the traveling salesperson problem where all distances between cities are either 1 or 2. Our main results are two approximation algorithms for (1,2)-TSP, one with approximation factor 8/7 and run time O(n^3) and the other having an approximation guarantee of 7/6 and run time O(n^{2.5}). The 8/7-approximation matches the best known approximation factor for (1,2)-TSP, due to Berman and Karpinski (SODA 2006), but considerably improves the previous best run time of O(n^9). Thus, ours is the first improvement for the (1,2)-TSP problem in more than 10 years. The algorithm is based on combining three copies of a minimum-cost cycle cover of the input graph together with a relaxed version of a minimum weight matching, which allows using "half-edges". The resulting multigraph is then edge-colored with four colors so that each color class yields a collection of vertex-disjoint paths. The paths from one color class can then be extended to an 8/7-approximate traveling salesperson tour. Our algorithm, and in particular its analysis, is simpler than the previously best 8/7-approximation. The 7/6-approximation algorithm is similar and even simpler, and has the advantage of not using Hartvigsen's complicated algorithm for computing a minimum-cost triangle-free cycle cover.

BibTeX - Entry

@InProceedings{adamaszek_et_al:LIPIcs:2018:9013,
  author =	{Anna Adamaszek and Matthias Mnich and Katarzyna Paluch},
  title =	{{New Approximation Algorithms for (1,2)-TSP}},
  booktitle =	{45th International Colloquium on Automata, Languages, and  Programming (ICALP 2018)},
  pages =	{9:1--9:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-076-7},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{107},
  editor =	{Ioannis Chatzigiannakis and Christos Kaklamanis and D{\'a}niel Marx and Donald Sannella},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2018/9013},
  URN =		{urn:nbn:de:0030-drops-90133},
  doi =		{10.4230/LIPIcs.ICALP.2018.9},
  annote =	{Keywords: Approximation algorithms, traveling salesperson problem, cycle cover}
}

Keywords: Approximation algorithms, traveling salesperson problem, cycle cover
Collection: 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)
Issue Date: 2018
Date of publication: 04.07.2018


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