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

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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 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.

## References

- 1.Aggarwal, N., Garg, N., Gupta, S.: A 4/3-approximation for TSP on cubic 3-edge-connected graphs. CoRR abs/1101.5586 (2011)Google Scholar
- 2.Balinski, M.L.: Integer programming: methods, uses, computation. Manag. Sci.
**12**, 253–313 (1965)CrossRefzbMATHMathSciNetGoogle Scholar - 3.Berman, P., Karpinski, M.: 8/7-approximation algorithm for (1,2)-TSP. In: Proceedings of the 17th ACM-SIAM Symposium on Discrete Algorithms, pp. 641–648 (2006)Google Scholar
- 4.Bläser, M., Shankar Ram, L.: An improved approximation algorithm for TSP with distances one and two. In: Liskiewicz, M., Reischuk, R. (eds.) Fundamentals of Computation Theory, 15th International Symposium, FCT 2005, Lecture Notes in Computer Science, vol. 3623, pp. 504–515. Springer (2005)Google Scholar
- 5.Boyd, S., Carr, R.: Finding low cost TSP and 2-matching solutions using certain half-integer subtour vertices. Discrete Optim.
**8**, 525–539 (2011). Prior version available at http://www.site.uottawa.ca/sylvia/recentpapers/halftri.pdf. Accessed 27 June 2011 - 6.Boyd, S., Sitters, R., van der Ster, S., Stougie, L.: The traveling salesman problem on cubic and subcubic graphs. Math. Program.
**144**(1–2), 227–245 (2014). A preliminary version appeared in IPCO 2011Google Scholar - 7.Christofides, N.: Worst case analysis of a new heuristic for the traveling salesman problem. Report 388, Graduate School of Industrial Administration, Carnegie-Mellon University, Pittsburgh, PA (1976)Google Scholar
- 8.Dantzig, G., Fulkerson, R., Johnson, S.: Solution of a large-scale traveling-salesman problem. Oper. Res.
**2**, 393–410 (1954)MathSciNetGoogle Scholar - 9.Gamarnik, D., Lewenstein, M., Sviridenko, M.: An improved upper bound for the TSP in cubic 3-edge-connected graphs. Oper. Res. Lett.
**33**(5), 467–474 (2005)Google Scholar - 10.Goemans, M.X.: Worst-case comparison of valid inequalities for the TSP. Math. Program.
**69**, 335–349 (1995)zbMATHMathSciNetGoogle Scholar - 11.Goemans, M.X., Bertsimas, D.J.: Survivable networks, linear programming relaxations, and the parsimonious property. Math. Program.
**60**, 145–166 (1990)CrossRefMathSciNetGoogle Scholar - 12.IBM ILOG CPLEX 12.1 (2009)Google Scholar
- 13.McKay, B.D.: Practical graph isomorphism. Congr. Numerantium
**30**, 45–97 (1981)MathSciNetGoogle Scholar - 14.Mnich, M., Mömke, T.: Improved integrality gap upper bounds for TSP with distances one and two. CoRR abs/1312.2502 (2013)Google Scholar
- 15.Mömke, T., Svensson, O.: Approximating graphic TSP by matchings. In: Proceedings of the 52nd Annual IEEE Symposium on Foundations of Computer Science, pp. 560–569 (2011)Google Scholar
- 16.Mucha, M.: \(\frac{13}{9}\)-approximation for graphic TSP. Theory Comput. Syst., 1–18 (2012). A preliminary version appeared in STACS 2012Google Scholar
- 17.Oveis Gharan, S., Saberi, A., Singh, M.: A randomized rounding approach to the traveling salesman problem. In: Proceedings of the 52nd Annual IEEE Symposium on Foundations of Computer Science, pp. 550–559 (2011)Google Scholar
- 18.Papadimitriou, C.H., Yannakakis, M.: The traveling salesman problem with distances one and two. Math. Oper. Res.
**18**, 1–11 (1993)CrossRefzbMATHMathSciNetGoogle Scholar - 19.Qian, J., Schalekamp, F., Williamson, D.P., van Zuylen, A.: On the integrality gap of the subtour LP for the 1,2-TSP. In: LATIN 2012: Theoretical Informatics, 10th Latin American Symposium, Lecture Notes in Computer Science, vol. 7256, pp. 606–617 (2012)Google Scholar
- 20.Schalekamp, F., Williamson, D.P., van Zuylen, A.: 2-matchings, the traveling salesman problem, and the subtour LP: A proof of the Boyd-Carr conjecture. Math. Oper. Res.
**39**(2), 403–417 (2014). A preliminary version appeared in SODA 2012Google Scholar - 21.Sebő, A., Vygen, J.: Shorter tours by nicer ears: 7/5-approximation for graphic TSP, 3/2 for the path version, and 4/3 for two-edge-connected subgraphs. CoRR abs/1201.1870 (2012)Google Scholar
- 22.Shmoys, D.B., Williamson, D.P.: Analyzing the Held–Karp TSP bound: a monotonicity property with application. Inf. Process. Lett.
**35**, 281–285 (1990)CrossRefzbMATHMathSciNetGoogle Scholar - 23.Williamson, D.P.: Analysis of the Held–Karp heuristic for the traveling salesman problem. Master’s thesis, MIT, Cambridge, MA (1990). Also appears as Tech Report MIT/LCS/TR-479Google Scholar
- 24.Wolsey, L.A.: Heuristic analysis, linear programming and branch and bound. Math. Program. Study
**13**, 121–134 (1980)CrossRefzbMATHMathSciNetGoogle Scholar