Skip to main content
SpringerLink
Log in

Approximation Algorithms for Semidefinite Packing Problems with Applications to Maxcut and Graph Coloring

  • Conference paper
Integer Programming and Combinatorial Optimization (IPCO 2005)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 3509))

Abstract

We describe the semidefinite analog of the vector packing problem, and show that the semidefinite programming relaxations for Maxcut [10] and graph coloring [17] are in this class of problems. We extend a method of Bienstock and Iyengar [5] which was based on ideas from Nesterov [25] to design an algorithm for computing ε-approximate solutions for this class of semidefinite programs. Our algorithm is in the spirit of Klein and Lu [18], and decreases the dependence of the run-time on ε from ε − − 2 to ε − − 1. For sparse graphs, our method is faster than the best specialized interior point methods. A significant feature of our method is that it treats both the Maxcut and the graph coloring problem in a unified manner.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Alizadeh, F.: Interior point methods in semidefinite programming with applications to combinatorial optimization. SIAM J. Optim. 5(1), 13–51 (1995)

    Article  MATH  MathSciNet  Google Scholar 

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

    Google Scholar 

  3. Benson, S.J., Ye, Y., Zhang, X.: Solving large-scale sparse semidefinite programs for combinatorial optimization. SIAM J. Optim. 10(2), 443–461 (2000) (electronic)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bienstock, D.: Potential function methods for approximately solving linear programming problems: theory and practice. International Series in Operations Research & Management Science, vol. 53. Kluwer Academic Publishers, Boston (2002)

    MATH  Google Scholar 

  5. Bienstock, D., Iyengar, G.: Solving fractional packing problems in \({O}^{*}(\frac{1}{\epsilon})\) iterations. In: Proceedings of the 36th Annual ACM Symposium on Theory of Computing, pp. 146–155 (2004)

    Google Scholar 

  6. Burer, S., Monteiro, R.D.C.: A projected gradient algorithm for solving the maxcut SDP relaxation. Optim. Methods Softw. 15(3-4), 175–200 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  7. Fleischer, L.: Fast approximation algorithms for fractional covering problems with box constraint. In: Proceedings of the 15th ACM-SIAM Symposium on Discrete Algorithms (2004)

    Google Scholar 

  8. Fleischer, L.K.: Approximating fractional multicommodity flow independent of the number of commodities. SIAM J. Discrete Math. 13(4), 505–520 (2000) (electronic)

    Article  MATH  MathSciNet  Google Scholar 

  9. Garg, N., Konemann, 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. Goemans, M.X., Williamson, D.P.: 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 

  11. Golub, G.H., Van Loan, C.F.: Matrix computations. Johns Hopkins Series in the Mathematical Sciences, vol. 3. Johns Hopkins University Press, Baltimore (1983)

    MATH  Google Scholar 

  12. Grigoriadis, M.D., Khachiyan, L.G.: Fast approximation schemes for convex programs with many blocks and coupling constraints. SIAM Journal on Optimization 4(1), 86–107 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  13. Grigoriadis, M.D., Khachiyan, L.G.: An exponential-function reduction method for block angular convex programs. Networks 26, 59–68 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  14. Grötschel, M., Lovász, L., Schrijver, A.: Polynomial algorithms for perfect graphs. In: Topics on perfect graphs. North-Holland Math. Stud, vol. 88, pp. 325–356. North-Holland, Amsterdam (1984)

    Chapter  Google Scholar 

  15. Hochbruck, M., Lubich, C.: On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  16. Iyengar, G., Phillips, D.J., Stein, C.: Approximation algorithms for semidefinite packing problems with applications to maxcut and graph coloring. Technical Report TR-2004-06, Computational Optimization Research Center, Columbia University (2004), Available at http://www.corc.ieor.columbia.edu/reports/techreports/tr-2004-06.pdf

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

    Article  MATH  MathSciNet  Google Scholar 

  18. Klein, P., Lu, H.-I.: Efficient approximation algorithms for semidefinite programs arising from MAX CUT and COLORING. In: Proceedings of the Twenty-eighth Annual ACM Symposium on the Theory of Computing, Philadelphia, PA, pp. 338–347. ACM, New York (1996)

    Google Scholar 

  19. Klein, P., Lu, H.-I.: Space-efficient approximation algorithms for MAXCUT and COLORING semidefinite programs. In: Chwa, K.-Y., Ibarra, O.H. (eds.) ISAAC 1998. LNCS, vol. 1533, pp. 387–396. Springer, Heidelberg (1998)

    Chapter  Google Scholar 

  20. Klein, P., Plotkin, S.A., Stein, C., Tardos, É.: Faster approximation algorithms for the unit capacity concurrent flow problem with applications to routing and finding sparse cuts. SIAM Journal on Computing 23(3), 466–487 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  21. Leighton, T., Makedon, F., Plotkin, S., Stein, C., Tardos, É., Tragoudas, S.: Fast approximation algorithms for multicommodity flow problems. Journal of Computer and System Sciences 50, 228–243 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  22. Lovász, L.: On the Shannon capacity of a graph. IEEE Trans. Inform. Theory 25(1), 1–7 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  23. Moler, C., Van Loan, C.: Nineteen dubious ways to compute the exponential of a matrix. SIAM Rev. 20(4), 801–836 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  24. Moler, C., Van Loan, C.: Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later. SIAM Rev. 45(1), 3–49 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  25. Nesterov, Y.: Smooth minimization of nonsmooth functions. Technical report, CORE DP (2003)

    Google Scholar 

  26. Nesterov, Y., Nemirovski, A.: Interior-point polynomial algorithms in convex programming. SIAM Studies in Applied Mathematics, vol. 13. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (1994)

    MATH  Google Scholar 

  27. Plotkin, S., Shmoys, D.B., Tardos, E.: Fast approximation algorithms for fractional packing and covering problems. Mathematics of Operations Research 20, 257–301 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  28. Radzik, T.: Fast deterministic approximation for the multicommodity flow problem. In: Proceedings of the 6th ACM-SIAM Symposium on Discrete Algorithms, pp. 486–496 (1995)

    Google Scholar 

  29. Shahrokhi, F., Matula, D.W.: The maximum concurrent flow problem. Journal of the ACM 37, 318–334 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  30. Skutella, M.: Convex quadratic and semidefinite programming relaxations in scheduling. Journal of the ACM 48(2), 206–242 (2001)

    Article  MathSciNet  Google Scholar 

  31. van den Eshof, J., Hochbruck, M.: Preconditioning lanczos approximations to the matrix exponential. To appear in SIAM J. of Sci. Comp. (2004)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of IEOR, Columbia University, New York, NY

    G. Iyengar, D. J. Phillips & C. Stein

Authors
  1. G. Iyengar
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. D. J. Phillips
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. C. Stein
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Institut für Informatik, Universität zu Köln,  

    Michael Jünger

  2. Fakultät für Mathematik, Otto-von-Guericke Universität Magdeburg, Universitätsplatz 2, 39106, Magdeburg, Germany

    Volker Kaibel

Rights and permissions

Reprints and permissions

Copyright information

© 2005 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Iyengar, G., Phillips, D.J., Stein, C. (2005). Approximation Algorithms for Semidefinite Packing Problems with Applications to Maxcut and Graph Coloring. In: Jünger, M., Kaibel, V. (eds) Integer Programming and Combinatorial Optimization. IPCO 2005. Lecture Notes in Computer Science, vol 3509. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11496915_12

Download citation

  • DOI: https://doi.org/10.1007/11496915_12

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-26199-5

  • Online ISBN: 978-3-540-32102-6

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation