Mathematical Programming

, Volume 159, Issue 1–2, pp 1–29 | Cite as

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

  • Joseph Cheriyan
  • Zhihan Gao
  • Konstantinos Georgiou
  • Sahil Singla
Full Length Paper Series A

Abstract

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).

Keywords

Asymmetric TSP Sherali–Adams hierarchy Integrality ratios 

Mathematics Subject Classification

90C05 90C27 

References

  1. 1.
    Anari, N., Gharan, S.O.: Effective-resistance-reducing flows and Asymmetric TSP. CoRR, abs/1411.4613 (2014)Google Scholar
  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)MathSciNetCrossRefMATHGoogle 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)Google Scholar
  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)Google Scholar
  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)Google Scholar
  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)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle 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)Google Scholar
  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)MathSciNetCrossRefMATHGoogle 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)Google Scholar
  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)Google Scholar
  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)Google Scholar
  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)Google Scholar
  15. 15.
    Lasserre, J.B.: An explicit equivalent positive semidefinite program for nonlinear 0–1 programs. SIAM J. Optim. 12(3), 756–769 (2002)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Papadimitriou, C.H., Vempala, S.: On the approximability of the Traveling Salesman Problem. Combinatorica 26(1), 101–120 (2006)MathSciNetCrossRefMATHGoogle 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)CrossRefGoogle 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)MathSciNetCrossRefMATHGoogle 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)Google Scholar
  22. 22.
    Watson, T.: Lift-and-project integrality gaps for the Traveling Salesperson Problem. Electron. Colloq. Comput. Complex. (ECCC) 18, 97 (2011)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2015

Authors and Affiliations

  • Joseph Cheriyan
    • 1
  • Zhihan Gao
    • 1
  • Konstantinos Georgiou
    • 1
  • Sahil Singla
    • 2
  1. 1.Department of Combinatorics and OptimizationUniversity of WaterlooWaterlooCanada
  2. 2.School of Computer ScienceCarnegie Mellon UniversityPittsburghUSA

Personalised recommendations