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DOI: 10.4230/LIPIcs.ESA.2019.56
URN: urn:nbn:de:0030-drops-111772
URL: http://drops.dagstuhl.de/opus/volltexte/2019/11177/
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Haddadan, Arash ; Newman, Alantha

Towards Improving Christofides Algorithm for Half-Integer TSP

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LIPIcs-ESA-2019-56.pdf (0.5 MB)


Abstract

We study the traveling salesman problem (TSP) in the case when the objective function of the subtour linear programming relaxation is minimized by a half-cycle point: x_e in {0,1/2,1} where the half-edges form a 2-factor and the 1-edges form a perfect matching. Such points are sufficient to resolve half-integer TSP in general and they have been conjectured to demonstrate the largest integrality gap for the subtour relaxation. For half-cycle points, the best-known approximation guarantee is 3/2 due to Christofides' famous algorithm. Proving an integrality gap of alpha for the subtour relaxation is equivalent to showing that alpha x can be written as a convex combination of tours, where x is any feasible solution for this relaxation. To beat Christofides' bound, our goal is to show that (3/2 - epsilon)x can be written as a convex combination of tours for some positive constant epsilon. Let y_e = 3/2-epsilon when x_e = 1 and y_e = 3/4 when x_e = 1/2. As a first step towards this goal, our main result is to show that y can be written as a convex combination of tours. In other words, we show that we can save on 1-edges, which has several applications. Among them, it gives an alternative algorithm for the recently studied uniform cover problem. Our main new technique is a procedure to glue tours over proper 3-edge cuts that are tight with respect to x, thus reducing the problem to a base case in which such cuts do not occur.

BibTeX - Entry

@InProceedings{haddadan_et_al:LIPIcs:2019:11177,
  author =	{Arash Haddadan and Alantha Newman},
  title =	{{Towards Improving Christofides Algorithm for Half-Integer TSP}},
  booktitle =	{27th Annual European Symposium on Algorithms (ESA 2019)},
  pages =	{56:1--56:12},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-124-5},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{144},
  editor =	{Michael A. Bender and Ola Svensson and Grzegorz Herman},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2019/11177},
  URN =		{urn:nbn:de:0030-drops-111772},
  doi =		{10.4230/LIPIcs.ESA.2019.56},
  annote =	{Keywords: Traveling salesman problem, subtour elimination relaxation, integrality gap, gluing subtours}
}

Keywords: Traveling salesman problem, subtour elimination relaxation, integrality gap, gluing subtours
Seminar: 27th Annual European Symposium on Algorithms (ESA 2019)
Issue Date: 2019
Date of publication: 06.09.2019


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