On integrality ratios for asymmetric TSP in the Sherali–Adams hierarchy


We study the Asymmetric Traveling Salesman Problem (ATSP), and our focus is on negative results in the framework of the Sherali–Adams (SA) Lift and Project method. Our main result pertains to the standard linear programming (LP) relaxation of ATSP, due to Dantzig, Fulkerson, and Johnson. For any fixed integer \(t\ge 0\) and small \(\epsilon \), \(0<\epsilon \ll {1}\), there exists a digraph G on \(\nu =\nu (t,\epsilon )=O(t/\epsilon )\) vertices such that the integrality ratio for level t of the SA system starting with the standard LP on G is \({\ge } 1+\frac{1-\epsilon }{2t+3} \approx \frac{4}{3}, \frac{6}{5}, \frac{8}{7}, \ldots \). Thus, in terms of the input size, the result holds for any \(t = 0,1,\ldots ,{\varTheta }(\nu )\) levels. Our key contribution is to identify a structural property of digraphs that allows us to construct fractional feasible solutions for any level t of the SA system starting from the standard LP. Our hard instances are simple and satisfy the structural property. There is a further relaxation of the standard LP called the balanced LP, and our methods simplify considerably when the starting LP for the SA system is the balanced LP; in particular, the relevant structural property (of digraphs) simplifies such that it is satisfied by the digraphs given by the well-known construction of Charikar, Goemans and Karloff (CGK). Consequently, the CGK digraphs serve as hard instances, and we obtain an integrality ratio of \(1 +\frac{1-\epsilon }{t+1}\) for any level t of the SA system, where \(0<\epsilon \ll {1}\) and the number of vertices is \(\nu (t,\epsilon )=O((t/\epsilon )^{(t/\epsilon )})\). Also, our results for the standard LP extend to the path ATSP (find a min cost Hamiltonian dipath from a given source vertex to a given sink vertex).

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7


  1. 1.

    Although the term integrality ratio is used in two different senses—one refers to an instance, the other to a relaxation (i.e., all instances)—the context will resolve the ambiguity.

  2. 2.

    By the integral hull we mean the convex hull of the zero-one solutions that are feasible for the original relaxation.


  1. 1.

    Anari, N., Gharan, S.O.: Effective-resistance-reducing flows and Asymmetric TSP. CoRR, abs/1411.4613 (2014)

  2. 2.

    Arora, S., Bollobás, B., Lovász, L., Tourlakis, I.: Proving integrality gaps without knowing the linear program. Theory Comput. 2(1), 19–51 (2006)

    MathSciNet  Article  MATH  Google Scholar 

  3. 3.

    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: Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2010, Austin, TX. 17–19 Jan 2010. pp. 379–389. SIAM (2010)

  4. 4.

    Au, Y.-H., Tunçel, L.: Complexity analyses of Bienstock-Zuckerberg and Lasserre relaxations on the matching and stable set polytopes. In: Günlük, O., Woeginger, G.J. (eds). IPCO, volume 6655 of Lecture Notes in Computer Science, pp. 14–26. Springer, Berlin (2011)

  5. 5.

    Benabbas, S., Chan, S.O., Georgiou, K., Magen, A.: Tight gaps for vertex cover in the Sherali–Adams SDP hierarchy. In: IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science, FSTTCS, volume 13 of LIPIcs, pp. 41–54 (2011)

  6. 6.

    Carr, R., Vempala, S.: On the Held–Karp relaxation for the asymmetric and symmetric traveling salesman problems. Math. Program. 100(3), 569–587 (2004)

    MathSciNet  Article  MATH  Google Scholar 

  7. 7.

    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)

    MathSciNet  Article  MATH  Google Scholar 

  8. 8.

    Charikar, M., Makarychev, K., Makarychev, Y.: Integrality gaps for Sherali–Adams relaxations. In: Proceedings of the 41st Annual ACM Symposium on Theory of Computing, STOC ’09, pp. 283–292. ACM, New York, NY (2009)

  9. 9.

    Cheung, K.K.H.: On Lovász–Schrijver lift-and-project procedures on the Dantzig–Fulkerson–Johnson relaxation of the TSP. SIAM J. Optim. 16(2), 380–399 (2005)

    MathSciNet  Article  MATH  Google Scholar 

  10. 10.

    Chlamtáč, E., Tulsiani, M.: Convex relaxations and integrality gaps. In: Anjos, M.F., Lasserre, J.B. (eds.) Handbook on Semidefinite: Conic and Polynomial Optimization, volume 166 of International Series in Operations Research and Management Science, pp. 139–169. Springer, US (2012)

    Google Scholar 

  11. 11.

    de la Vega W.F., Kenyon-Mathieu, C.: Linear programming relaxations of maxcut. In: Bansal, N., Pruhs, K., Stein, C. (eds). SODA’07 Proceedings of the Eighteenth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 53–61. ACM Press, New York (2007)

  12. 12.

    Elliott-Magwood, P.: The integrality gap of the Asymmetric Travelling Salesman Problem. PhD thesis, Department of Mathematics and Statistics, University of Ottawa (2008)

  13. 13.

    Karpinski, M., Lampis, M., Schmied, R.: New inapproximability bounds for TSP. In: Cai, L., Cheng, S.-W., Lam, T.W. (eds). ISAAC: Algorithms and Computation—24th International Symposium, ISAAC 2013, Hong Kong, China, December 16–18, 2013, Proceedings, volume 8283 of Lecture Notes in Computer Science, pp. 568–578. Springer, Berlin (2013)

  14. 14.

    Lampis, M.: Improved inapproximability for TSP. In: APPROX-RANDOM: Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques—15th International Workshop, APPROX 2012, and 16th International Workshop, RANDOM 2012, Cambridge, MA. August 15–17, 2012. Proceedings, volume 7408 of Lecture Notes in Computer Science, pp. 243–253. Springer (2012)

  15. 15.

    Lasserre, J.B.: An explicit equivalent positive semidefinite program for nonlinear 0–1 programs. SIAM J. Optim. 12(3), 756–769 (2002)

    MathSciNet  Article  MATH  Google Scholar 

  16. 16.

    Laurent, M.: A comparison of the Sherali–Adams, Lovász–Schrijver, and Lasserre relaxations for 0–1 programming. Math. Oper. Res. 28(3), 470–496 (2003)

    MathSciNet  Article  MATH  Google Scholar 

  17. 17.

    Lovász, L., Schrijver, A.: Cones of matrices and set-functions and 0–1 optimization. SIAM J. Optim. 1(2), 166–190 (1991)

    MathSciNet  Article  MATH  Google Scholar 

  18. 18.

    Papadimitriou, C.H., Vempala, S.: On the approximability of the Traveling Salesman Problem. Combinatorica 26(1), 101–120 (2006)

    MathSciNet  Article  MATH  Google Scholar 

  19. 19.

    Roberti, R., Toth, P.: Models and algorithms for the Asymmetric Traveling Salesman Problem: an experimental comparison. EURO J. Transp. Logist. 1, 113–133 (2012)

    Article  Google Scholar 

  20. 20.

    Sherali, H.D., Adams, W.P.: A hierarchy of relaxations between the continuous and convex hull representations for zero-one programming problems. SIAM J. Discrete Math. 3(3), 411–430 (1990)

    MathSciNet  Article  MATH  Google Scholar 

  21. 21.

    Tourlakis, I.: New lower bounds for Approximation Algorithms in the Lovasz–Schrijver Hierarchy. PhD thesis, Department of Computer Science, Princeton University (2006)

  22. 22.

    Watson, T.: Lift-and-project integrality gaps for the Traveling Salesperson Problem. Electron. Colloq. Comput. Complex. (ECCC) 18, 97 (2011)

    Google Scholar 

Download references


We thank a number of colleagues for useful discussions. We are grateful to Sylvia Boyd and Paul Elliott-Magwood for help with ATSP integrality gaps, and to Levent Tunçel for sharing his knowledge of the area.

Author information



Corresponding author

Correspondence to Konstantinos Georgiou.

Additional information

An extended abstract of this work appeared in the proceedings of the 40th International Colloquium on Automata, Languages, and Programming (ICALP 2013).

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Cheriyan, J., Gao, Z., Georgiou, K. et al. On integrality ratios for asymmetric TSP in the Sherali–Adams hierarchy. Math. Program. 159, 1–29 (2016). https://doi.org/10.1007/s10107-015-0947-5

Download citation


  • Asymmetric TSP
  • Sherali–Adams hierarchy
  • Integrality ratios

Mathematics Subject Classification

  • 90C05
  • 90C27