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Improved Approximation Guarantees through Higher Levels of SDP Hierarchies

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Approximation, Randomization and Combinatorial Optimization. Algorithms and Techniques (APPROX 2008, RANDOM 2008)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 5171))

Abstract

For every fixed γ ≥ 0, we give an algorithm that, given an n-vertex 3-uniform hypergraph containing an independent set of size γn, finds an independent set of size \(n^{\Omega(\gamma^2)}\). This improves upon a recent result of Chlamtac, which, for a fixed ε> 0, finds an independent set of size n ε in any 3-uniform hypergraph containing an independent set of size \((\frac12-\varepsilon)n\). The main feature of this algorithm is that, for fixed γ, it uses the Θ(1/γ 2)-level of a hierarchy of semidefinite programming (SDP) relaxations. On the other hand, we show that for at least one hierarchy which gives such a guarantee, 1/γ levels yield no non-trivial guarantee. Thus, this is a first SDP-based algorithm for which the approximation guarantee improves indefinitely as one uses progressively higher-level relaxations.

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References

  1. Alekhnovich, M., Arora, S., Tourlakis, I.: Towards strong nonapproximability results in the Lovász-Schrijver hierarchy. In: Proceedings of 37th Annual ACM Symposium on Theory of Computing, pp. 294–303 (2005)

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  3. Arora, S., Chlamtac, E., Charikar, M.: New approximation guarantee for chromatic number. In: Proceedings of 38th ACM Symposium of Theory of Computing, pp. 215–224 (2006)

    Google Scholar 

  4. Arora, S., Rao, S., Vazirani, U.: Expander flows, geometric embeddings and graph partitioning. In: Proceedings of the 36th Annual ACM Symposium on Theory of Computing, pp. 222–231 (2004)

    Google Scholar 

  5. Charikar, M., Makarychev, K., Makarychev, Y.: Integrality Gaps for Sherali-Adams Relaxations (manuscript)

    Google Scholar 

  6. Charikar, M., Makarychev, K., Makarychev, Y.: Near-Optimal Algorithms for Unique Games. In: Proceedings of the 38th Annual ACM Symposium on Theory of Computing, pp. 205–214 (2006)

    Google Scholar 

  7. Chlamtac, E.: Approximation Algorithms Using Hierarchies of Semidefinite Programming Relaxations. In: Proceedings of 48th IEEE Symposium on Foundations of Computer Science, pp. 691–701 (2007)

    Google Scholar 

  8. Klerk, E.D., Laurent, M., Parrilo, P.: A PTAS for the minimization of polynomials of fixed degree over the simplex. Theoretical Computer Science 361(2-3), 210–225 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  9. Feige, U., Krauthgamer, R.: The Probable Value of the Lovász-Schrijver Relaxations for Maximum Independent Set. SIAM Journal on Computing 32(2), 345–370 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  10. Feller, W.: An Introduction to Probability Theory and Its Applications, 3rd edn., vol. 1. John Wiley & Sons, Chichester (1968)

    MATH  Google Scholar 

  11. Georgiou, K., Magen, A., Pitassi, T., Tourlakis, I.: Integrality gaps of 2–o(1) for vertex cover in the Lovász-Schrijver hierarchy. In: Proceedings of 48th IEEE Symposium on Foundations of Computer Science, pp. 702–712 (2007)

    Google Scholar 

  12. Goemans, M., Williamson, D.: Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. Journal of the ACM 42(6), 1115–1145 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  13. Håstad, J.: Some optimal inapproximability results. Journal of the ACM 48(4), 798–859 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  14. Halperin, E., Nathaniel, R., Zwick, U.: Coloring k-colorable graphs using relatively small palettes. J. Algorithms 45(1), 72–90 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  15. Karger, D., Motwani, R., Sudan, M.: Approximate graph coloring by semidefinite programming. Journal of the ACM 45(2), 246–265 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  16. Karloff, H., Zwick, U.: A 7/8-approximation algorithm for max 3SAT? In: Proceedings of 38th IEEE Symposium on Foundations of Computer Science, pp. 406–415 (1997)

    Google Scholar 

  17. Khot, S.: On the power of unique 2-prover 1-round games. In: Proceedings of the 34th ACM Symposium on Theory of Computing, pp. 767–775 (2002)

    Google Scholar 

  18. Khot, S., Kindler, G., Mossel, E., O’Donnell, R.: Optimal inapproximability results for MAX-CUT and other 2-variable CSPs? SIAM Journal on Computing 37(1), 319–357 (2007); ECCC Report TR05-101 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  19. Krivelevich, M., Nathaniel, R., Sudakov, B.: Approximating coloring and maximum independent sets in 3-uniform hypergraphs. Journal of Algorithms 41(1), 99–113 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  20. Laurent, M.: A comparison of the Sherali-Adams, Lovász-Schrijver and Lasserre relaxations for 0-1 programming. Mathematics of Operations Research 28(3), 460–496 (2003)

    Article  MathSciNet  Google Scholar 

  21. Lasserre, J.B.: An explicit exact SDP relaxation for nonlinear 0-1 programs. In: Aardal, K., Gerards, B. (eds.) IPCO 2001. LNCS, vol. 2081, pp. 293–303. Springer, Heidelberg (2001)

    Google Scholar 

  22. Lovász, L., Schrijver, A.: Cones of matrices and set-functioins and 0-1 optimization. SIAM Journal on Optimization 1(2), 166–190 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  23. Nie, J., Schweighofer, M.: On the complexity of Putinar’s Positivstellensatz. Journal of Complexity 23(1), 135–150 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  24. Raghavendra, P.: Optimal Algorithms and Inapproximability Results For Every CSP? In: Proceedings of 30th ACM Symposium on Theory of Computing, pp. 245–254 (2008)

    Google Scholar 

  25. Schoenebeck, G., Trevisan, L., Tulsiani, M.: Tight integrality gaps for Lovász-Schrijver LP relaxations of Vertex Cover and Max Cut. In: Proceedings of 29th ACM Symposium on Theory of Computing, pp. 302–310 (2007)

    Google Scholar 

  26. 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)

    Article  MATH  MathSciNet  Google Scholar 

  27. Schoenebeck, G.: Personal communication

    Google Scholar 

  28. de la Vega, F.W., Kenyon-Mathieu, C.: Linear programming relaxations of maxcut. In: Proceedings of 18th ACM Symposium on Discrete Algorithms, pp. 53–61 (2007)

    Google Scholar 

  29. Zwick, U.: Computer assisted proof of optimal approximability results. In: Proceedings of 13th ACM Symposium on Discrete Algorithms, pp. 496–505 (2002)

    Google Scholar 

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Authors and Affiliations

  1. Department of Computer Science, Princeton University, Princeton, NJ 08544, USA

    Eden Chlamtac

  2. Departmen of Computer Science & Engineering, University of Washington, Box 352350, Seattle, WA 98195-2350, USA

    Gyanit Singh

Authors
  1. Eden Chlamtac
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  2. Gyanit Singh
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Editor information

Ashish Goel Klaus Jansen José D. P. Rolim Ronitt Rubinfeld

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© 2008 Springer-Verlag Berlin Heidelberg

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Chlamtac, E., Singh, G. (2008). Improved Approximation Guarantees through Higher Levels of SDP Hierarchies. In: Goel, A., Jansen, K., Rolim, J.D.P., Rubinfeld, R. (eds) Approximation, Randomization and Combinatorial Optimization. Algorithms and Techniques. APPROX RANDOM 2008 2008. Lecture Notes in Computer Science, vol 5171. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-85363-3_5

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  • DOI: https://doi.org/10.1007/978-3-540-85363-3_5

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-85362-6

  • Online ISBN: 978-3-540-85363-3

  • eBook Packages: Computer ScienceComputer Science (R0)

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