On the Integrality Gap of the Subtour LP for the 1,2-TSP

  • Jiawei Qian
  • Frans Schalekamp
  • David P. Williamson
  • Anke van Zuylen
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7256)

Abstract

In this paper, we study the integrality gap of the subtour LP relaxation for the traveling salesman problem in the special case when all edge costs are either 1 or 2. For the general case of symmetric costs that obey triangle inequality, a famous conjecture is that the integrality gap is 4/3. Little progress towards resolving this conjecture has been made in thirty years. We conjecture that when all edge costs cij ∈ {1,2}, the integrality gap is 10/9. We show that this conjecture is true when the optimal subtour LP solution has a certain structure. Under a weaker assumption, which is an analog of a recent conjecture by Schalekamp, Williamson and van Zuylen, we show that the integrality gap is at most 7/6. When we do not make any assumptions on the structure of the optimal subtour LP solution, we can show that inegrality gap is at most 19/15 ≈ 1.267 < 4/3; this is the first bound on the integrality gap of the subtour LP strictly less than 4/3 known for an interesting special case of the TSP.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Jiawei Qian
    • 1
  • Frans Schalekamp
    • 1
  • David P. Williamson
    • 1
  • Anke van Zuylen
    • 2
  1. 1.School of Operations Research and Information EngineeringCornell UniversityIthacaUSA
  2. 2.Department 1: Algorithms and ComplexityMax-Planck-Institut für InformatikSaarbrückenGermany

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