Skip to main content
SpringerLink
Log in
Book cover

Complementarity: Applications, Algorithms and Extensions pp 1–17Cite as

Approximating Maximum Stable Set and Minimum Graph Coloring Problems with the Positive Semidefinite Relaxation

  • Chapter

Part of the book series: Applied Optimization ((APOP,volume 50))

Abstract

We compute approximate solutions to the maximum stable set problem and the minimum graph coloring problem using a positive semidefinite relaxation. The positive semidefinite programs are solved using an implementation of the dual scaling algorithm that takes advantage of the sparsity inherent in most graphs and the structure inherent in the problem formulation. Prom the solution to the relaxation, we apply a randomized algorithm to find approximate maximum stable sets and a modification of a popular heuristic to find graph colorings. We obtained high quality answers for graphs with over 1000 vertices and over 6000 edges.

This is a preview of subscription content, 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   129.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   169.99
Price excludes VAT (USA)
  • Durable hardcover 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

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. F. Alizadeh. Combinatorial optimization with interior point methods and semidefinite matrices. PhD thesis, University of Minnesota, Minneapolis, MN, 1991.

    Google Scholar 

  2. K. M. Anstreicher and M. Fampa. A long-step path following algorithm for semidefinite programming problems. Working Paper, Department of Management Science, The University of Iowa, Iowa City, IA, 1996.

    Google Scholar 

  3. L. Babel. Finding maximum cliques in arbitrary and in special graphs. Computing, 46:321–341, 1991.

    Article  MathSciNet  MATH  Google Scholar 

  4. L. Babel and G. Tinhofer. A branch and bound algorithm for the maximum clique problem. J. of Global Optimization, 4, 1994.

    Google Scholar 

  5. Egon Balas and H. Samuelsson. A node covering algorithm. Naval Research Logistics Quarterly, 24(2):213–233, 1977.

    Article  MathSciNet  MATH  Google Scholar 

  6. Egon Balas and Jue Xue. Minimum weighted coloring of triangulated graphs, with application to maximum weight vertex packing and clique finding in arbitrary graphs. SIAM Journal on Computing, 20(2):209–221, April 1991.

    Article  MathSciNet  MATH  Google Scholar 

  7. Egon Balas and Chang Sung Yu. Finding a maximum clique in an arbitrary graph. SIAM Journal on Computing, 15(4):1054–1068, November 1986.

    Article  MathSciNet  MATH  Google Scholar 

  8. S. Benson, Y. Ye, and X. Zhang. Solving large-scale sparse semidef-inite programs for combinatorial optimization. Technical report, Department of Management Science, University of Iowa, Iowa City, IA 52242, USA, September 1997. To appear in SIAM J. of Optimization.

    Google Scholar 

  9. C. Berge. Graphs and Hypergraphs. North-Holland, Amsterdam, 1973.

    MATH  Google Scholar 

  10. D. Bertsimas and Y. Ye. Semidefinite relaxations, multivariate normal distributions, and order statistics. In D.-Z. Du and P.M. Parda-Los, editors, Handbook of Combinatorial Optimization, volume 3, pages 1–19. Springer Science+Business Media Dordrecht, 1998.

    Google Scholar 

  11. P. Briggs, K. Cooper, K. Kennedy, and L. Torczon. Coloring heuristics for register allocation. In ASCM Conference on Program Language Design and Implementation, pages 275–284. The Association for Computing Machinery, 1998.

    Google Scholar 

  12. R. Carrahan and P. M. Pardalos. An exact algorithm for the maximum clique problem. Operations Research Letters, 9:375–382, 1990.

    Article  Google Scholar 

  13. G.J. Chaitin, M. Auslander, A.K. Chandra, J. Cocke, M.E. Hopkins, and P. Markstein. Register allocation via coloring. Computer Languages, 6:47–57, 1981.

    Article  Google Scholar 

  14. Gregory J. Chaitin. Register allocation and spilling via graph coloring. SIGPLAN Notices (Proceedings of the SIGPLAN ′82 Symposium on Compiler Construction, Boston, Mass.), 17(6):98–101, June 1982.

    Article  Google Scholar 

  15. D. De Werra. An introduction to timetabling. European Journal of Operations Research, 19:151–162, 1985.

    Article  MATH  Google Scholar 

  16. DIMACS Center Web Page. The Second DIMACS Implementation Challenge: 1992–1993. http://dimacs.rutgers.edu/pub/challenge/graph/benchmarks/clique/.

  17. C. Priden, A. Hertz, and D. de Werra. An exact algorithm based on tabu search for finding a maximum independent set in a graph. Computers Operations Research, 17(5):375–382, 1990.

    Google Scholar 

  18. Andreas Gamst. Some lower bounds for a class of frequency assignment problems. IEEE Transactions of Vehicular Technology, 35(1):8–14, 1986.

    Article  Google Scholar 

  19. Michael R. Garey and David S. Johnson. Computers and Intractability: A Guide to the Theory of NP-Completeness. W.H Freeman, San Francisco, CA, 1979.

    MATH  Google Scholar 

  20. M. X. Goemans and D. P. Williamson. 878-approximation for MAX CUT and MAX 2SAT. In Proc. 26th ACM Symp. Theor. Computing, pages 422–431, 1994.

    Google Scholar 

  21. David S. Johnson. Approximation algorithms for combinatorial problems. Journal of Computer and System Sciences, 9:256–278, 1974.

    Article  MathSciNet  MATH  Google Scholar 

  22. David S. Johnson. Worst case behavior of graph coloring algorithms. In Proceedings of 5th Southeastern Conference on Combinatorics, Graph Theory, and Computing, pages 513–527. Utilitas Mathemat-ica, Winnipeg, Canada, 1974.

    Google Scholar 

  23. David Karger, Rajeev Motwani, and Madhu Sudan. Approximate graph coloring by semidefinite programming. Technical report, MIT, Cambridge, MA 52242, USA, 1994.

    Google Scholar 

  24. Jon Kleinberg and Michel X. Goemans. The Lovász theta function and a semidefinite programming relaxation of vertex cover. SIAM Journal on Discrete Mathematics, 11 (2):196–204, May 1998.

    Article  MathSciNet  MATH  Google Scholar 

  25. M. Kubale and B. Jackowski. A generalized implicit enumeration algorithm for graph coloring. Communications of the ACM, 28:412–418, 1985.

    Article  Google Scholar 

  26. M. Kubale and E. Kusz. Computational experience with implicit enumeration algorithms for graph coloring. In Proceedings of the WG’83 International Workshop on Graphtheoretic Concepts in Computer Science, pages 167–176, Linz, 1983. Trauner Verlag.

    Google Scholar 

  27. Carlo Mannino and Antonio Sassano. An exact algorithm for the maximum cardinality stable set problem. Networks, page (submitted), 1993. http://www.dimacs.rutgers.edu/pub/challenge/graph/contributed.

    Google Scholar 

  28. Anuj Mehrotra and Michael A. Trick. A column generation approach for graph coloring, http://mat.gsia.cmu.edu/trick.html, April 1995.

    Google Scholar 

  29. Craig A. Morgenstern and Harry D. Shapiro. Coloration neighborhood structures for general graph coloring. In Proceedings of the First Annual ACM-SIAM Symposium on Discrete Algorithms, San Francisco, Jan, 1990. Society for Industrial and Applied Mathematics, Philadelphia, 1990.

    Google Scholar 

  30. George L. Nemhauser and G. L. Sigismondi. A strong cutting plane /branch and bound algorithm for node packing. Journal of the Operational Research Society, 43(5), 1992.

    Google Scholar 

  31. George. L. Nemhauser and Les. E. Trotter, Jr. Vertex packings: Structural properties and algorithms. Mathematical Programming, 8(2):232–248, 1975.

    Article  MathSciNet  MATH  Google Scholar 

  32. Yu. E. Nesterov. Semidefinite relaxation and nonconvex quadratic optimization. Optimization Methods and Software, 9:141–160, 1998.

    Article  MathSciNet  MATH  Google Scholar 

  33. Yu. E. Nesterov and A. S. Nemirovskii. Interior Point Polynomial Methods in Convex Programming: Theory and Algorithms. SIAM Publications, SIAM, Philadelphia, 1993.

    Google Scholar 

  34. B. Pittel. On the probable behaviour of some algorithms for finding the stability number of a graph. Mathematical Proceedings of the Cambridge Philosophical Society, 92:511–526, 1982.

    Article  MathSciNet  MATH  Google Scholar 

  35. M. J. D. Powell and P. L. Toint. On the estimation of sparse Hessian matrices. SIAM Journal on Numerical Analysis, 16:1060–1074, 1979.

    Article  MathSciNet  MATH  Google Scholar 

  36. Proc. 4th IPCO Conference. Improved approximation algorithms for max k-cut and max bisection, 1995.

    Google Scholar 

  37. M. V. Ramana, L. Tunçel, and H. Wolkowicz. Strong duality for semidefinite programming. SIAM Journal on Optimization, 7:641–662, 1997.

    Article  MathSciNet  MATH  Google Scholar 

  38. T. Stephen and L. Tuncel. On a representation of the matching polytope via semidefinite liftings. Mathematics of Operations Research, 24(l):l-7, 1999.

    MathSciNet  Google Scholar 

  39. M. J. Todd. On search directions in interior-point methods for semidefinite programming. Technical Report 1205, School of Operations Research and Industrial Engineering, Cornell University, Itheca, NY 14853–3801, October 1997.

    Google Scholar 

  40. M. Trick. Graph coloring instances, http://mat.gsia.cmu.edu/COLOR/instances.html.

  41. L. Tuncel. On the slater condition for the sdp relaxations of noncon-vex sets. Technical Report CORR2000–13, Department of Combina-torica and Optimization, University of Waterloo, Waterloo, Ontario N2L3G1, Canada, February 2000.

    Google Scholar 

  42. L. Vandenberghe and S. Boyd. Semidefinite programming. SIAM Review, 38(l):49–95, 1996.

    Article  MathSciNet  MATH  Google Scholar 

  43. D. C. Wood. A technique for coloring a graph applicable to large scale time-tabling problems. The Computer Journal, 3:317–319, 1969.

    Article  Google Scholar 

  44. Y. Ye. Interior Point Algorithms: Theory and Analysis. Wiley-Interscience Series in Discrete Mathematics and Optimization. John Wiley & Sons, New York, 1997.

    Book  MATH  Google Scholar 

  45. Y. Ye. Approximating quadratic programming with bound and quadratic constraints. Mathematical Programming, 84:219–226, 1999.

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Division of Mathematics and Computer Science, Argonne National Laboratory, Argonne, IL, 60439, USA

    S. J. Benson

  2. Department of Management Sciences, The University of Iowa, Iowa City, Iowa, 52242, USA

    Y. Ye

Authors
  1. S. J. Benson
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Y. Ye
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Computer Sciences Department, University of Wisconsin, Madison, Wisconsin, USA

    Michael C. Ferris  & Olvi L. Mangasarian  & 

  2. Department of Mathematical Sciences, The Johns Hopkins University, Baltimore, Maryland, USA

    Jong-Shi Pang

Rights and permissions

Reprints and permissions

Copyright information

© 2001 Springer Science+Business Media Dordrecht

About this chapter

Cite this chapter

Benson, S.J., Ye, Y. (2001). Approximating Maximum Stable Set and Minimum Graph Coloring Problems with the Positive Semidefinite Relaxation. In: Ferris, M.C., Mangasarian, O.L., Pang, JS. (eds) Complementarity: Applications, Algorithms and Extensions. Applied Optimization, vol 50. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-3279-5_1

Download citation

  • DOI: https://doi.org/10.1007/978-1-4757-3279-5_1

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4419-4847-2

  • Online ISBN: 978-1-4757-3279-5

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation