Lift-and-Project Methods for Set Cover and Knapsack

  • Eden Chlamtáč
  • Zachary Friggstad
  • Konstantinos Georgiou
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8037)


We study the applicability of lift-and-project methods to the Set Cover and Knapsack problems. Inspired by recent work of Karlin, Mathieu, and Nguyen [IPCO 2011], who examined this connection for Knapsack, we consider the applicability and limitations of these methods for Set Cover, as well as extending extending the existing results for Knapsack.

For the Set Cover problem, Cygan, Kowalik, and Wykurz [IPL 2009] gave sub-exponential-time approximation algorithms with approximation ratios better than ln n. We present a very simple combinatorial algorithm which has nearly the same time-approximation tradeoff as the algorithm of Cygan et al. We then adapt this to an LP-based algorithm using the LP hierarchy of Lovász and Schrijver. However, our approach involves the trick of “lifting the objective function”. We show that this trick is essential, by demonstrating an integrality gap of (1 − ε)ln n at level Ω(n) of the stronger LP hierarchy of Sherali and Adams (when the objective function is not lifted).

Finally, we show that the SDP hierarchy of Lovász and Schrijver (LS + ) reduces the integrality gap for Knapsack to (1 + ε) at level O(1). This stands in contrast to Set Cover (where the work of Aleknovich, Arora, and Tourlakis [STOC 2005] rules out any improvement using LS + ), and extends the work of Karlin et al., who demonstrated such an improvement only for the more powerful SDP hierarchy of Lasserre. Our LS +  based rounding and analysis are quite different from theirs (in particular, not relying on the decomposition theorem they prove for the Lasserre hierarchy), and to the best of our knowledge represents the first explicit demonstration of such a reduction in the integrality gap of LS +  relaxations after a constant number of rounds.


Set Cover Sub-exponential Algorithms Approximation Algorithms Lift-and-Project Methods Knapsack 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Alekhnovich, M., Arora, S., Tourlakis, I.: Towards strong nonapproximability results in the Lovász-Schrijver hierarchy. In: Proceedings of ACM Symposium on Theory of Computing, pp. 294–303 (2005)Google Scholar
  2. 2.
    Barak, B., Raghavendra, P., Steurer, D.: Rounding semidefinite programming hierarchies via global correlation. In: Proceedings of IEEE Symposium on Foundations of Computer Science, pp. 472–481 (2011)Google Scholar
  3. 3.
    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, Heidelberg (2012)CrossRefGoogle Scholar
  4. 4.
    Chvátal, V.: A greedy heuristic for the set-covering problem. Mathematics of Operations Research 4(3), 233–235 (1979), doi:10.2307/3689577MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Cygan, M., Kowalik, L., Wykurz, M.: Exponential-time approximation of weighted set cover. Inf. Process. Lett. 109(16), 957–961 (2009)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Feige, U.: A threshold of ln n for approximating set cover. J. ACM 45(4), 634–652 (1998)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Guruswami, V., Sinop, A.K.: Lasserre hierarchy, higher eigenvalues, and approximation schemes for graph partitioning and quadratic integer programming with PSD objectives. In: Proceedings of IEEE Symposium on Foundations of Computer Science, pp. 482–491 (October 2011)Google Scholar
  8. 8.
    Ibarra, O.H., Kim, C.E.: Fast approximation algorithms for the knapsack and sum of subset problems. J. ACM 22(4), 463–468 (1975)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Impagliazzo, R., Paturi, R.: On the complexity of k-SAT. J. Comput. Syst. Sci. 62(2), 367–375 (2001)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Johnson, D.S.: Approximation algorithms for combinatorial problems. J. Comput. Syst. Sci. 9(3), 256–278 (1974)MATHCrossRefGoogle Scholar
  11. 11.
    Karlin, A.R., Mathieu, C., Nguyen, C.T.: Integrality gaps of linear and semi-definite programming relaxations for knapsack. In: Günlük, O., Woeginger, G.J. (eds.) IPCO 2011. LNCS, vol. 6655, pp. 301–314. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  12. 12.
    Karp, R.M.: Reducibility among combinatorial problems. In: Complexity of Computer Computations, pp. 85–103 (1972)Google 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.
    Lovász, L.: On the ratio of optimal integral and fractional covers. Discrete Mathematics 13(4), 383–390 (1975)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Lovász, L., Schrijver, A.: Cones of matrices and set-functions and 0-1 optimization. SIAM J. Optim. 1(2), 166–190 (1991)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Moshkovitz, D.: The projection games conjecture and the NP-hardness of ln n-approximating set-cover. In: Gupta, A., Jansen, K., Rolim, J., Servedio, R. (eds.) APPROX 2012 and RANDOM 2012. LNCS, vol. 7408, pp. 276–287. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  17. 17.
    Raghavendra, P.: Optimal algorithms and inapproximability results for every csp? In: Proceedings of ACM Symposium on Theory of Computing, pp. 245–254 (2008)Google Scholar
  18. 18.
    Raghavendra, P., Tan, N.: Approximating csps with global cardinality constraints using sdp hierarchies. In: Proceedings of ACM-SIAM Symposium on Discrete Algorithms, pp. 373–387. SIAM (2012)Google Scholar
  19. 19.
    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

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Eden Chlamtáč
    • 1
  • Zachary Friggstad
    • 2
  • Konstantinos Georgiou
    • 2
  1. 1.Department of Computer ScienceBen Gurion UniversityBeer ShevaIsrael
  2. 2.Department of Combinatorics and OptimizationUniversity of WaterlooWaterlooCanada

Personalised recommendations