On Integrality Ratios for Asymmetric TSP in the Sherali-Adams Hierarchy

  • Joseph Cheriyan
  • Zhihan Gao
  • Konstantinos Georgiou
  • Sahil Singla
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7965)

Abstract

We study the ATSP (Asymmetric Traveling Salesman Problem), 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 LP (linear programming) relaxation of ATSP, due to Dantzig, Fulkerson, and Johnson. For any fixed integer t ≥ 0 and small ε, 0 < ε ≪ 1, there exists a digraph G on ν = ν(t,ε) = O(t/ε) vertices such that the integrality ratio for level t of the SA system starting with the standard LP on G is \(\ge 1+\frac{1-\epsilon}{2t+3} \approx \frac43, \frac65, \frac87, \dots\). Thus, in terms of the input size, the result holds for any t = 0,1,…,Θ(ν) 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 < ε ≪ 1 and the number of vertices is ν(t,ε) = O((t/ε)(t/ε)).

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).

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Arora, S., Bollobás, B., Lovász, L., Tourlakis, I.: Proving integrality gaps without knowing the linear program. Theory of Computing 2(1), 19–51 (2006)MathSciNetCrossRefGoogle Scholar
  2. 2.
    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: Proc. ACM–SIAM SODA 2010, pp. 379–389. SIAM (2010)Google Scholar
  3. 3.
    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 2011. LNCS, vol. 6655, pp. 14–26. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  4. 4.
    Benabbas, S., Chan, S.O., Georgiou, K., Magen, A.: Tight gaps for vertex cover in the Sherali-Adams SDP hierarchy. In: Proc. FSTTCS 2011. LIPIcs, vol. 13, pp. 41–54 (2011)Google Scholar
  5. 5.
    Carr, R., Vempala, S.: On the Held-Karp relaxation for the asymmetric and symmetric traveling salesman problems. Math. Program. 100(3), 569–587 (2004)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    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)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Charikar, M., Makarychev, K., Makarychev, Y.: Integrality gaps for Sherali-Adams relaxations. In: Proc. ACM STOC 2009, New York, NY, USA, pp. 283–292 (2009)Google Scholar
  8. 8.
    Cheung, K.K.H.: On Lovász–Schrijver lift-and-project procedures on the Dantzig–Fulkerson–Johnson relaxation of the TSP. SIAM Journal on Optimization 16(2), 380–399 (2005)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Chlamtáč, E., Tulsiani, M.: Convex relaxations and integrality gaps. In: Anjos, M.F., Lasserre, J.B. (eds.) Handbook on Semidefinite, Conic and Polynomial Optimization. International Series in Operations Research & Management Science, vol. 166, pp. 139–169. Springer, US (2012)CrossRefGoogle Scholar
  10. 10.
    de la Vega, W.F., Kenyon-Mathieu, C.: Linear programming relaxations of maxcut. In: Proc. ACM–SIAM SODA 2007, pp. 53–61. ACM Press (2007)Google Scholar
  11. 11.
    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
  12. 12.
    Lampis, M.: Improved inapproximability for TSP. In: Gupta, A., Jansen, K., Rolim, J., Servedio, R. (eds.) APPROX/RANDOM 2012. LNCS, vol. 7408, pp. 243–253. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  13. 13.
    Lasserre, J.B.: An explicit equivalent positive semidefinite program for nonlinear 0-1 programs. SIAM Journal on Optimization 12(3), 756–769 (2002)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    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)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Lovász, L., Schrijver, A.: Cones of matrices and set-functions and 0-1 optimization. SIAM Journal on Optim. 1(2), 166–190 (1991)MATHCrossRefGoogle Scholar
  16. 16.
    Papadimitriou, C.H., Vempala, S.: On the approximability of the traveling salesman problem. Combinatorica 26(1), 101–120 (2006)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Roberti, R., Toth, P.: Models and algorithms for the Asymmetric Traveling Salesman Problem: an experimental comparison. EURO Journal on Transportation and Logistics 1, 113–133 (2012)CrossRefGoogle Scholar
  18. 18.
    Sherali, H.D., Adams, W.P.: A hierarchy of relaxations between the continuous and convex hull representations for zero-one programming problems. SIAM Journal on Discrete Mathematics 3(3), 411–430 (1990)MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Tourlakis, I.: New lower bounds for Approximation Algorithms in the Lovasz-Schrijver Hierarchy. PhD thesis, Department of Computer Science, Princeton University (2006)Google Scholar
  20. 20.
    Watson, T.: Lift-and-project integrality gaps for the Traveling Salesperson Problem. Electronic Colloquium on Computational Complexity (ECCC) 18, 97 (2011)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Joseph Cheriyan
    • 1
  • Zhihan Gao
    • 1
  • Konstantinos Georgiou
    • 1
  • Sahil Singla
    • 1
  1. 1.University of WaterlooWaterlooCanada

Personalised recommendations