2-Matchings, the Traveling Salesman Problem, and the Subtour LP: A Proof of the Boyd-Carr Conjecture

In this paper,we study the integrality gap of the subtour LP relaxation for thetraveling salesman problem in the special case when all edge costs are either 1 or 2. Forthe general case of symmetric costs that obey triangle inequality, a famous conjectureis that the integrality gap is 4/3. Little progress towards resolving this conjecturehas been made in 30years. We conjecture that when all edge costs ci j ∈ {1, 2}, theintegrality gap is 10/9. We show that this conjecture is true when the optimal subtourLP solution has a certain structure. Under a weaker assumption, which is an analog ofa recent conjecture by Schalekamp et al., we show that the integrality gap is at most7/6. When we do not make any assumptions on the structure of the optimal subtour LP A preliminary version of this paper appeared in LATIN 2012: Theoretical Informatics [19]. Some resultsin the current paper are stronger, using the same techniques as in [19]. The first author was supported inpart by NSF grant CCF-1115256. This work was carried out in part while the third author was onsabbatical at TU Berlin, and the fourth author was at Max-Planck-Institut für Informatik in Saarbrücken,Germany. The third author was supported in part by the Berlin Mathematical School, the Alexander vonHumboldt Foundation, and NSF Grant CCF-1115256. J. QianBank of America Merrill Lynch, Chicago, IL 60661, USAe-mail: [email protected] F. Schalekamp · A. van Zuylen(B)Department of Mathematics, College of William and Mary, Williamsburg, VA 23185, USAe-mail: [email protected] F. Schalekampe-mail: [email protected] D. P. WilliamsonSchool of Operations Research and Information Engineering, Cornell University,Ithaca, NY 14853, USAe-mail: [email protected]