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

  • S. J. Benson
  • Y. Ye
Part of the Applied Optimization book series (APOP, volume 50)


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.


Stable Set Independent Set Maximum Clique Graph Coloring Positive Semidefinite Relaxation 


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  1. [1]
    F. Alizadeh. Combinatorial optimization with interior point methods and semidefinite matrices. PhD thesis, University of Minnesota, Minneapolis, MN, 1991.Google Scholar
  2. [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. [3]
    L. Babel. Finding maximum cliques in arbitrary and in special graphs. Computing, 46:321–341, 1991.MathSciNetzbMATHCrossRefGoogle Scholar
  4. [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. [5]
    Egon Balas and H. Samuelsson. A node covering algorithm. Naval Research Logistics Quarterly, 24(2):213–233, 1977.MathSciNetzbMATHCrossRefGoogle Scholar
  6. [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.MathSciNetzbMATHCrossRefGoogle Scholar
  7. [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.MathSciNetzbMATHCrossRefGoogle Scholar
  8. [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. [9]
    C. Berge. Graphs and Hypergraphs. North-Holland, Amsterdam, 1973.zbMATHGoogle Scholar
  10. [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. [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. [12]
    R. Carrahan and P. M. Pardalos. An exact algorithm for the maximum clique problem. Operations Research Letters, 9:375–382, 1990.CrossRefGoogle Scholar
  13. [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.CrossRefGoogle Scholar
  14. [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.CrossRefGoogle Scholar
  15. [15]
    D. De Werra. An introduction to timetabling. European Journal of Operations Research, 19:151–162, 1985.zbMATHCrossRefGoogle Scholar
  16. [16]
    DIMACS Center Web Page. The Second DIMACS Implementation Challenge: 1992–1993.
  17. [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. [18]
    Andreas Gamst. Some lower bounds for a class of frequency assignment problems. IEEE Transactions of Vehicular Technology, 35(1):8–14, 1986.CrossRefGoogle Scholar
  19. [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.zbMATHGoogle Scholar
  20. [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. [21]
    David S. Johnson. Approximation algorithms for combinatorial problems. Journal of Computer and System Sciences, 9:256–278, 1974.MathSciNetzbMATHCrossRefGoogle Scholar
  22. [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. [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. [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.MathSciNetzbMATHCrossRefGoogle Scholar
  25. [25]
    M. Kubale and B. Jackowski. A generalized implicit enumeration algorithm for graph coloring. Communications of the ACM, 28:412–418, 1985.CrossRefGoogle Scholar
  26. [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. [27]
    Carlo Mannino and Antonio Sassano. An exact algorithm for the maximum cardinality stable set problem. Networks, page (submitted), 1993. Scholar
  28. [28]
    Anuj Mehrotra and Michael A. Trick. A column generation approach for graph coloring,, April 1995.Google Scholar
  29. [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. [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. [31]
    George. L. Nemhauser and Les. E. Trotter, Jr. Vertex packings: Structural properties and algorithms. Mathematical Programming, 8(2):232–248, 1975.MathSciNetzbMATHCrossRefGoogle Scholar
  32. [32]
    Yu. E. Nesterov. Semidefinite relaxation and nonconvex quadratic optimization. Optimization Methods and Software, 9:141–160, 1998.MathSciNetzbMATHCrossRefGoogle Scholar
  33. [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. [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.MathSciNetzbMATHCrossRefGoogle Scholar
  35. [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.MathSciNetzbMATHCrossRefGoogle Scholar
  36. [36]
    Proc. 4th IPCO Conference. Improved approximation algorithms for max k-cut and max bisection, 1995.Google Scholar
  37. [37]
    M. V. Ramana, L. Tunçel, and H. Wolkowicz. Strong duality for semidefinite programming. SIAM Journal on Optimization, 7:641–662, 1997.MathSciNetzbMATHCrossRefGoogle Scholar
  38. [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.MathSciNetGoogle Scholar
  39. [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. [40]
    M. Trick. Graph coloring instances,
  41. [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. [42]
    L. Vandenberghe and S. Boyd. Semidefinite programming. SIAM Review, 38(l):49–95, 1996.MathSciNetzbMATHCrossRefGoogle Scholar
  43. [43]
    D. C. Wood. A technique for coloring a graph applicable to large scale time-tabling problems. The Computer Journal, 3:317–319, 1969.CrossRefGoogle Scholar
  44. [44]
    Y. Ye. Interior Point Algorithms: Theory and Analysis. Wiley-Interscience Series in Discrete Mathematics and Optimization. John Wiley & Sons, New York, 1997.zbMATHCrossRefGoogle Scholar
  45. [45]
    Y. Ye. Approximating quadratic programming with bound and quadratic constraints. Mathematical Programming, 84:219–226, 1999.MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2001

Authors and Affiliations

  • S. J. Benson
    • 1
  • Y. Ye
    • 2
  1. 1.Division of Mathematics and Computer ScienceArgonne National LaboratoryArgonneUSA
  2. 2.Department of Management SciencesThe University of IowaIowa CityUSA

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