On the integrality gap of the subtour LP for the 1,2-TSP

  • Authors:

  • Jiawei Qian;Frans Schalekamp;David P. Williamson;Anke van Zuylen

  • Affiliations:

  • School of Operations Research and Information Engineering, Cornell University, Ithaca, NY;School of Operations Research and Information Engineering, Cornell University, Ithaca, NY;School of Operations Research and Information Engineering, Cornell University, Ithaca, NY;Department 1: Algorithms and Complexity, Max-Planck-Institut für Informatik, Saarbrücken, Germany

  • Venue:
  • LATIN'12 Proceedings of the 10th Latin American international conference on Theoretical Informatics
  • Year:
  • 2012

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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