Algorithmica

, Volume 69, Issue 3, pp 605–618 | Cite as

An SDP Primal-Dual Algorithm for Approximating the Lovász-Theta Function

Article

Abstract

The Lovász ϑ-function (Lovász in IEEE Trans. Inf. Theory, 25:1–7, 1979) of a graph G=(V,E) can be defined as the maximum of the sum of the entries of a positive semidefinite matrix X, whose trace Tr(X) equals 1, and Xij=0 whenever {i,j}∈E. This function appears as a subroutine for many algorithms for graph problems such as maximum independent set and maximum clique. We apply Arora and Kale’s primal-dual method for SDP to design an algorithm to approximate the ϑ-function within an additive error of δ>0, which runs in time \(O(\frac{\vartheta ^{2} n^{2}}{\delta^{2}} \log n \cdot M_{e})\), where ϑ=ϑ(G) and Me=O(n3) is the time for a matrix exponentiation operation. It follows that for perfect graphs G, our primal-dual method computes ϑ(G) exactly in time O(ϑ2n5logn).

Moreover, our techniques generalize to the weighted Lovász ϑ-function, and both the maximum independent set weight and the maximum clique weight for vertex weighted perfect graphs can be approximated within a factor of (1+ϵ) in time O(ϵ−2n5logn).

Keywords

Approximation algorithms Primal-dual SDP methods Lovász-Theta function Perfect graphs Maximum independent sets 

References

  1. 1.
    Arora, S., Hazan, E., Kale, S.: The multiplicative weights update method: a meta-algorithm and applications. http://www.cs.princeton.edu/~ehazan/papers/MWsurvey.pdf
  2. 2.
    Arora, S., Hazan, E., Kale, S.: \({O}(\sqrt{\log n})\) approximation to sparsest cut in \(\tilde{O}(n^{2})\) time. In: Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science, pp. 238–247 (2004) CrossRefGoogle Scholar
  3. 3.
    Arora, S., Kale, S.: A combinatorial, primal-dual approach to semidefinite programs. In: Proceedings of the 39th Annual ACM Symposium on Theory of Computing, pp. 227–236 (2007) Google Scholar
  4. 4.
    Alizadeh, F.: Interior point methods in semidefinite programming with applications to combinatorial optimization. SIAM J. Optim. 5(1), 13–51 (1995) CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Berge, C.: Färbung von Graphen, deren sämtliche bzw. deren ungerade Kreise starr sind. Wiss. Z., Martin-Luther-Univ. Halle-Wittenb., Math.-Nat.wiss. Reihe 10, 114 (1961) Google Scholar
  6. 6.
    Chudnovsky, M., Cornuéjols, G., Liu, X., Seymour, P., Vusković, K.: Recognizing |Berge graphs. Combinatorica 25, 143–186 (2005) CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Chudnovsky, M., Robertson, N., Seymour, P., Thomas, R.: The strong perfect graph theorem. Ann. Math. 164, 51–229 (2006) CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Eisenbrand, F., Funke, S., Garg, N., Könemann, J.: A combinatorial algorithm for computing a maximum independent set in a t-perfect graph. In: Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 517–522 (2003) Google Scholar
  9. 9.
    Garg, N., Könemann, J.: Faster and simpler algorithms for multicommodity flow and other fractional packing problems. In: Proceedings of the 39th Annual Symposium on Foundations of Computer Science, pp. 300–309 (1998) Google Scholar
  10. 10.
    Grotchel, L., Lovasz, L., Schrijver, A.: Polynomial algorithms for perfect graphs. In: Annals of Discrete Mathematics, pp. 325–356 (1984) Google Scholar
  11. 11.
    Grotchel, L., Lovasz, L., Schrijver, A.: Geometric Algorithms and Combinatorial Optimization. Springer, Berlin (1987) Google Scholar
  12. 12.
    Golumbic, M.C.: Algorithmic Graph Theory and Perfect Graphs. Academic Press, New York (1980) MATHGoogle Scholar
  13. 13.
    Golub, G.H., van Loan, C.F.: Matrix Computations, 3rd edn. Johns Hopkins University Press, Baltimore (1996) MATHGoogle Scholar
  14. 14.
    Iyengar, G., Phillips, D.J., Stein, C.: Approximating semidefinite packing programs. SIAM J. Optim. 21(1), 231–268 (2011) CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Klein, P.N., Lu, H.-I.: Efficient approximation algorithms for semidefinite programs arising from max cut and coloring. In: Proceedings of the 28th Annual ACM Symposium on Theory of Computing, pp. 338–347 (1996) Google Scholar
  16. 16.
    Karger, D.R., Motwani, R., Sudan, M.: Approximate graph coloring by semidefinite programming. J. ACM 45(2), 246–265 (1998) CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Knuth, D.E.: The sandwich theorem. Electron. J. Comb. 1 (1994) Google Scholar
  18. 18.
    Koufogiannakis, C., Young, N.E.: Beating simplex for fractional packing and covering linear programs. In: Proceedings of the 48th Annual IEEE Symposium on Foundations of Computer Science, pp. 494–504 (2007) Google Scholar
  19. 19.
    Lovász, L.: On the Shannon capacity of a graph. IEEE Trans. Inf. Theory 25, 1–7 (1979) CrossRefMATHGoogle Scholar
  20. 20.
    Nayakkankuppam, M.V., Overton, M.L.: Primal-dual interior-point methods for semidefinite programming: numerical experience with block-diagonal problems. In: Proceedings of the 1996 IEEE International Symposium on Computer-Aided Control System Design, pp. 235–239 (1996) Google Scholar
  21. 21.
    Pan, V.Y., Chen, Z.Q.: The complexity of the matrix eigenproblem. In: Proceedings of the Thirty-First Annual ACM Symposium on Theory of Computing, STOC ’99, pp. 507–516. ACM, New York (1999) Google Scholar
  22. 22.
    Plotkin, S.A., Shmoys, D.B., Tardos, É.: Fast approximation algorithms for fractional packing and covering problems. In: Proceedings of the 32nd Annual Symposium on Foundations of Computer Science, pp. 495–504 (1991) CrossRefGoogle Scholar
  23. 23.
    Seymour, P.: How the proof of the strong perfect graph conjecture was found. Gaz. Math. 109, 69–83 (2006) MATHMathSciNetGoogle Scholar
  24. 24.
    Shannon, C.E.: The zero-error capacity of a noisy channel. IRE Trans. Inf. Theory 2(3), 8–19 (1956) CrossRefMathSciNetGoogle Scholar
  25. 25.
    Vazirani, V.V.: Primal-dual schema based approximation algorithms (abstract). In: Computing and Combinatorics, pp. 650–652 (1995) CrossRefGoogle Scholar
  26. 26.
    vanden Eshof, J., Hochbruck, M.: Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • T.-H. Hubert Chan
    • 1
  • Kevin L. Chang
    • 3
  • Rajiv Raman
    • 2
  1. 1.Department of Computer ScienceUniversity of Hong KongPokfulamHong Kong
  2. 2.TCS Innovation LabsTRDDCPuneIndia
  3. 3.Mountain ViewUSA

Personalised recommendations