Mathematical Programming

, Volume 150, Issue 1, pp 131–151

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

  • Jiawei Qian
  • Frans Schalekamp
  • David P. Williamson
  • Anke van Zuylen
Full Length Paper Series B

DOI: 10.1007/s10107-014-0835-4

Cite this article as:
Qian, J., Schalekamp, F., Williamson, D.P. et al. Math. Program. (2015) 150: 131. doi:10.1007/s10107-014-0835-4

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 30 years. We conjecture that when all edge costs \(c_{ij}\in \{1,2\}\), 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 et al., 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 integrality gap is at most 5/4; 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. We show computationally that the integrality gap is at most 10/9 for all instances with at most 12 cities.

Keywords

Traveling salesman problem Subtour elimination  Linear programming Integrality gap 

Mathematics Subject Classification

90C05 90C27 05C70 

Copyright information

© Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2014

Authors and Affiliations

  • Jiawei Qian
    • 1
  • Frans Schalekamp
    • 2
  • David P. Williamson
    • 3
  • Anke van Zuylen
    • 2
  1. 1.Bank of America Merrill LynchChicagoUSA
  2. 2.Department of MathematicsCollege of William and MaryWilliamsburgUSA
  3. 3.School of Operations Research and Information EngineeringCornell UniversityIthacaUSA

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