Semidefinite and Linear Programming Integrality Gaps for Scheduling Identical Machines

  • Adam Kurpisz
  • Monaldo Mastrolilli
  • Claire Mathieu
  • Tobias Mömke
  • Victor Verdugo
  • Andreas Wiese
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9682)

Abstract

Sherali-Adams [25] and Lovász-Schrijver [21] developed systematic procedures to strengthen a relaxation known as lift-and-project methods. They have been proven to be a strong tool for developing approximation algorithms, matching the best relaxations known for problems like Max-Cut and Sparsest-Cut. In this work we provide lower bounds for these hierarchies when applied over the configuration LP for the problem of scheduling identical machines to minimize the makespan. First we show that the configuration LP has an integrality gap of at least \(1024\text {/}1023\) by providing a family of instances with 15 different job sizes. Then we show that for any integer n there is an instance with n jobs in this family such that after \(\varOmega (n)\) rounds of the Sherali-Adams (\(\text {SA}\)) or the Lovász-Schrijver (\(\text {LS}_+\)) hierarchy the integrality gap remains at least \(1024\text {/}1023\).

References

  1. 1.
    Alekhnovich, M., Arora, S., Tourlakis, I.: Towards strong nonapproximability results in the Lovász-Schrijver hierarchy. In: STOC, pp. 294–303 (2005)Google Scholar
  2. 2.
    Alon, N., Azar, Y., Woeginger, G.J., Yadid, T.: Approximation schemes for scheduling. In: SODA, pp. 493–500 (1997)Google Scholar
  3. 3.
    Arora, S., Bollobás, B., Lovász, L., Tourlakis, I.: Proving integrality gaps without knowing the linear program. Theor. Comput. 2, 19–51 (2006)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Bansal, N.: Approximating independent sets in sparse graphs. In: SODA, pp. 1–8 (2015)Google Scholar
  5. 5.
    Bansal, N., Srinivasan, A., Svensson, O.: Lift-and-round to improve weighted completion time on unrelated machines. CoRR, abs/1511.07826 (2015)Google Scholar
  6. 6.
    Buresh-Oppenheim, J., Galesi, N., Hoory, S., Magen, A., Pitassi, T.: Rank bounds and integrality gaps for cutting planes procedures. Theor. Comput. 2, 65–90 (2006)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Charikar, M.: On semidefinite programming relaxations for graph coloring and vertex cover. In: SODA, pp. 616–620 (2002)Google Scholar
  8. 8.
    Chlamtáč, E., Friggstad, Z., Georgiou, K.: Lift-and-project methods for set cover and Knapsack. In: Dehne, F., Solis-Oba, R., Sack, J.-R. (eds.) WADS 2013. LNCS, vol. 8037, pp. 256–267. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  9. 9.
    Chlamtac, E., Krauthgamer, R., Raghavendra, P.: Approximating sparsest cut in graphs of bounded treewidth. In: Serna, M., Shaltiel, R., Jansen, K., Rolim, J. (eds.) APPROX 2010. LNCS, vol. 6302, pp. 124–137. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  10. 10.
    Chlamtac, E., Tulsiani, M.: Convex relaxations and integrality gaps. In: Anjos, M.F., Lasserre, J.B. (eds.) Handbook on Semidefinite, Conic and Polynomial Optimization, pp. 139–169. Springer, New York (2012)CrossRefGoogle Scholar
  11. 11.
    Fernandez de la Vega, W., Mathieu, C.: Linear programming relaxations of maxcut. In: SODA, pp. 53–61 (2007)Google Scholar
  12. 12.
    Feige, U., Krauthgamer, R.: The probable value of the Lovász-Schrijver relaxations for maximum independent set. SIAM J. Comput. 32, 345–370 (2003)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Garey, M.R., Johnson, D.S.: “Strong” NP-completeness results: motivation, examples, and implications. J. ACM 25, 499–508 (1978)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Georgiou, K., Magen, A., Pitassi, T., Tourlakis, I.: Integrality gaps of 2-o(1) for vertex cover SDPs in the Lovász-Schrijver hierarchy. SIAM J. Comput. 39, 3553–3570 (2010)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Gvozdenovic, N., Laurent, M.: The operator \(\psi \) for the chromatic number of a graph. SIAM J. Optim. 19, 572–591 (2008)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Hochbaum, D., Shmoys, D.: Using dual approximation algorithms for scheduling problems theoretical and practical results. J. ACM 34, 144–162 (1987)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Kurpisz, Adam, Leppänen, S., Mastrolilli, M.: A Lasserre lower bound for the min-sum single machine scheduling problem. In: Bansal, N., Finocchi, I. (eds.) ESA 2015. LNCS, vol. 9294, pp. 853–864. Springer, Heidelberg (2015). doi:10.1007/978-3-662-48350-3_71 CrossRefGoogle Scholar
  18. 18.
    Lasserre, J.: Global optimization with polynomials and the problem of moments. SIAM J. Optim. 11, 796–817 (2001)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Laurent, M.: A comparison of the Sherali-Adams, Lovász-Schrijver, and Lasserre relaxations for 0–1 programming. Math. Oper. Res. 28, 470–496 (2003)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Levey, E., Rothvoss, T.: A Lasserre-based \((1+\varepsilon )\)-approximation for \(\text{P}m | p_j=1, \text{ prec } | \text{ C }_{\max }\). CoRR, abs/1509.07808 (2015)Google Scholar
  21. 21.
    Lovász, L., Schrijver, A.: Cones of matrices and set-functions and 0–1 optimization. SIAM J. Optim. 1, 166–190 (1991)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Parrilo, P.: Semidefinite programming relaxations for semialgebraic problems. Math. Program. 96, 293–320 (2003)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Rothvoß, T.: The Lasserre hierarchy in approximation algorithms. Lecture notes for the MAPSP (2013)Google Scholar
  24. 24.
    Schoenebeck, G., Trevisan, L., Tulsiani, M.: Tight integrality gaps for Lovász-Schrijver LP relaxations of vertex cover and max cut. In: Proceedings of the Thirty-Ninth Annual ACM Symposium on Theory of Computing, pp. 302–310 (2007)Google Scholar
  25. 25.
    Sherali, H., Adams, W.: A hierarchy of relaxations between the continuous and convex hull representations for zero-one programming problems. SIAM J. Discrete Math. 3, 411–430 (1990)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Verschae, J., Wiese, A.: On the configuration-LP for scheduling on unrelated machines. J. Sched. 17, 371–383 (2014)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Williamson, D., Shmoys, D.: The Design of Approximation Algorithms. Cambridge University Press, New York (2011)CrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Adam Kurpisz
    • 1
  • Monaldo Mastrolilli
    • 1
  • Claire Mathieu
    • 2
  • Tobias Mömke
    • 3
  • Victor Verdugo
    • 2
    • 4
  • Andreas Wiese
    • 5
  1. 1.Dalle Molle Institute for Artificial Intelligence ResearchMannoSwitzerland
  2. 2.Department of Computer ScienceCNRS UMR 8548, École normale supérieureParisFrance
  3. 3.Department of Computer ScienceSaarland UniversitySaarbrückenGermany
  4. 4.Department of Industrial EngineeringUniversidad de ChileSantiagoChile
  5. 5.Max Planck Institute for InformaticsSaarbrückenGermany

Personalised recommendations