Improved integrality gap upper bounds for traveling salesperson problems with distances one and two

Matthias Mnich, Tobias Momke

Research output: Contribution to journalArticleAcademicpeer-review

Abstract

We prove new integrality gap upper bounds for the traveling salesperson problem with distances one and two ((1,2)-TSP). We obtain these bounds by investigating the structure of the support graph of optimal solutions to the subtour elimination linear programming relaxation, with one additional cutting plane inequality.

For undirected (1,2)-TSP, our main results are as follows:

All instances have an integrality gap of at most 5/4.

Instances admitting half-integral solutions have integrality gap at most 7/6.

Instances admitting subcubic solutions Of cost at most the order of the instance have integrality gap at most 10/9, even without the cutting plane. This bound is tight, and holds in particular for basic solutions in the fractional 2-matching polytope with cost at most the order of the instance.

For directed (1,2)-TSP instances we show an integrality gap upper bound of 3/2 for general instances, and of 5/4 for instances admitting half-integral solutions.

We prove these bounds by providing local search algorithms that, in polynomial time, find 2-matchings with few components in the support of the solution. We show that the run times of our algorithms cannot be considerably improved under standard complexity-theoretic assumptions: we show that finding improved TSP solutions via local search is intractable for edge change parameterized by the size of the neighborhoods even for instances with distances one and two; this strengthens a result of Daniel Marx. (C) 2017 Elsevier B.V. All rights reserved.

Original languageEnglish
Pages (from-to)436-457
Number of pages22
JournalEuropean Journal of Operational Research
Volume266
Issue number2
DOIs
Publication statusPublished - 16 Apr 2018

Keywords

  • Traveling salesman
  • Integrality gap
  • Subtour elimination
  • SALESMAN PROBLEM
  • APPROXIMATION ALGORITHM
  • WEIGHTS ZERO
  • SUBTOUR LP
  • TSP

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