Mathematical Programming

, Volume 150, Issue 1, pp 131–151 | Cite as

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

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 

Notes

Acknowledgments

We thank Sylvia Boyd for useful and encouraging discussions. We thank two anonymous referees for helpful comments and suggestions.

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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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