Mathematical Programming

, Volume 141, Issue 1–2, pp 549–559 | Cite as

A simple LP relaxation for the asymmetric traveling salesman problem

Full Length Paper Series A


It is a long-standing open question in combinatorial optimization whether the integrality gap of the subtour linear program relaxation (subtour LP) for the asymmetric traveling salesman problem (ATSP) is a constant. The study on the structure of this linear program is important and extensive. In this paper, we give a new and simpler LP relaxation for the ATSP. Our linear program consists of a single type of constraints that combine both the subtour elimination and the degree constraints in the traditional subtour LP. As a result, we obtain a much simpler relaxation. In particular, it is shown that the extreme solutions of our program have at most 2n − 2 non-zero variables, improving the bound 3n − 2, proved by Vempala and Yannakakis, for the ones obtained by the subtour LP. Nevertheless, the integrality gap of the new linear program is larger than that of the traditional subtour LP by at most a constant factor.


Subtour elimination linear program relaxation Extreme solutions Traveling salesman problem 

Mathematics Subject Classification



Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Asadpour, A., Goemans, M.X., Madry, A., Gharan, S.O., Saberi, A.: An o(log n/ log log n)-approximation algorithm for the asymmetric traveling salesman problem. In: SODA, pp. 379–389 (2010)Google Scholar
  2. 2.
    Carr R.D., Vempala S.: On the held-karp relaxation for the asymmetric and symmetric traveling salesman problems. Math. Program. 100(3), 569–587 (2004)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Charikar M., Goemans M.X., Karloff H.J.: On the integrality ratio for the asymmetric traveling salesman problem. Math. Oper. Res. 31(2), 245–252 (2006)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Cook W.: Combinatorial Optimization. Wiley, New York (1998)MATHGoogle Scholar
  5. 5.
    Dantzig G., Fulkerson D., Johnson S.: Solutions of a large-scale traveling salesman problem. Oper. Res. 2, 393–410 (1954)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Frieze A.M., Galbiati G., Maffioli F.: On the worst-case performance of some algorithms for the asymmetric traveling salesman problem. Networks 12(1), 23–39 (1982)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Gutin G., Punnen A., Barvinok A., Gimadi E.K., Serdyukov A.I.: The Traveling Salesman Problem and Its Variations. Springer, Berlin (2002)MATHGoogle Scholar
  8. 8.
    Held M., Karp R.M.: The traveling-salesman problem and minimum spanning trees. Oper. Res. 18(6), 1138–1162 (1970)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Hoffman, A.: Some recent applications of the theory of linear inequalities to extremal combinatorial analysis. In: Proceeding of Symposium in Applied Mathematics, Amer. Math. Soc. pp. 113–127 (1960)Google Scholar
  10. 10.
    Jain K.: A factor 2 approximation algorithm for the generalized steiner network problem. Combinatorica 21(1), 39–60 (2001)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Lau L.C., Naor J., Salavatipour M.R., Singh M.: Survivable network design with degree or order constraints. SIAM J. Comput. 39(3), 1062–1087 (2009)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Lawler E.: The Traveling Salesman Problem: A Guided Tour of Combinatorial Optimization. Wiley-Interscience Series in Discrete Mathematics and Optimization. Wiley, New York (1985)Google Scholar
  13. 13.
    Vempala, S., Yannakakis, M.: A convex relaxation for the asymmetric tsp. In: SODA, pp. 975–976 (1999)Google Scholar

Copyright information

© Springer and Mathematical Optimization Society 2012

Authors and Affiliations

  1. 1.Managerial Economics and Decision Sciences, Kellogg School of ManagementNorthwestern UniversityEvanstonUSA

Personalised recommendations