Skip to main content
Springer Nature Link
Log in

Semidefinite programming in combinatorial optimization

  • Published:
Mathematical Programming Submit manuscript

Abstract

We discuss the use of semidefinite programming for combinatorial optimization problems. The main topics covered include (i) the Lovász theta function and its applications to stable sets, perfect graphs, and coding theory, (ii) the automatic generation of strong valid inequalities, (iii) the maximum cut problem and related problems, and (iv) the embedding of finite metric spaces and its relationship to the sparsest cut problem.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

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

    Article  MATH  MathSciNet  Google Scholar 

  2. F. Alizadeh, J.-P. Haeberly and M. Overton, Complementarity and nondegeneracy in semidefinite programming, Technical Report, Courant Institute of Mathematical Sciences, 1995; also in:Mathematical Programming, to appear.

  3. N. Alon, Eigenvalues and expanders,Combinatorica 6 (1986) 83–96.

    Article  MATH  MathSciNet  Google Scholar 

  4. N. Alon and N. Kahale, Approximating the independence number via theϑ function, Manuscript, 1994.

  5. N. Alon and V. Milman,λ 1, isoperimetric inequalities for graphs and superconcentrators,Journal of Combinatorial Theory B 38 (1985) 73–88.

    Article  MATH  MathSciNet  Google Scholar 

  6. Y. Aumann and Y. Rabani, An O (logk) approximate min-cut max-flow theorem and approximation algorithm,SIAM Journal on Computing, to appear.

  7. E. Balas, S. Ceria and G. Cornuéjols, A lift-and-project cutting plane algorithm for mixed 0–1 programs,Mathematical Programming 58 (1993) 295–324.

    Article  MathSciNet  Google Scholar 

  8. F. Barahona and A. Mahjoub, On the cut polytope,Mathematical Programming 36 (1986) 157–173.

    Article  MATH  MathSciNet  Google Scholar 

  9. A.I. Barvinok, Problems of distance geometry and convex properties of quadratic maps,Discrete and Computational Geometry 13 (1995) 189–202.

    Article  MATH  MathSciNet  Google Scholar 

  10. J. Bourgain, On Lipschitz embedding of finite metric spaces in Hilbert space,Israel Journal of Mathematics 52 (1985) 46–52.

    MATH  MathSciNet  Google Scholar 

  11. A. Brouwer and W. Haemers, Association schemes, in: R. Graham, M. Grötschel and L. Lovász, eds.,Handbook of Combinatorics (North-Holland, Amsterdam, 1995) 747–771.

    Google Scholar 

  12. B. Chor and M. Sudan, A geometric approach to betweenness, in:Proceedings of the 3rd European Symposium on Algorithms, Lecture Notes in Computer Science, Vol. 979 (Springer, Berlin, 1995) 227–237.

    Google Scholar 

  13. C. Delorme and S. Poljak, Combinatorial properties and the complexity of a max-cut approximation,European Journal of Combinatorics 14 (1993) 313–333.

    Article  MATH  MathSciNet  Google Scholar 

  14. C. Delorme and S. Poljak, Laplacian eigenvalues and the maximum cut problem,Mathematical Programming 62 (1993) 557–574.

    Article  MathSciNet  Google Scholar 

  15. P. Delsarte, An algebraic approach to the association schemes of coding theory,Philips Research Reports, Supplement 10 (1973).

  16. M. Deza and M. Laurent,Geometry of Cuts and Metrics, Algorithms and Combinatorics (Springer, Berlin, 1997).

    Google Scholar 

  17. U. Feige, Randomized graph products, chromatic numbers, and the Lovászϑ-function, in:Proceedings of the 27th ACM Symposium on Theory of Computing (1995) 635–640.

  18. U. Feige and M.X. Goemans, Approximating the value of two prover proof systems, with applications to MAX 2SAT and MAX DICUT, in:Proceedings of the 3rd Israel Symposium on Theory Comput. and Syst. (1995) 182–189.

  19. A. Frieze and M. Jerrum, Improved approximation algorithms for MAXk-CUT and MAX BISECTION,Algorithmica 18 (1997) 67–81.

    Article  MATH  MathSciNet  Google Scholar 

  20. M.X. Goemans and D.P. Williamson, Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming,J. ACM 42 (1995) 1115–1145; A preliminary version entitled “.878-Approximation algorithms for MAXCUT and MAX2SAT” has appeared inProceedings 26th ACM Symposium on Theory of Computing (1994) 422–431.

    Article  MATH  MathSciNet  Google Scholar 

  21. M. Golumbic,Algorithmic Graph Theory and Perfect Graphs (Academic Press, New York, 1980).

    MATH  Google Scholar 

  22. M. Grötschel, L. Lovász and A. Schrijver, The ellipsoid method and its consequences in combinatorial optimization,Combinatorica 1 (1981) 169–197.

    Article  MATH  MathSciNet  Google Scholar 

  23. M. Grötschel, L. Lovász and A. Schrijver, Relaxations of vertex packing,Journal of Combinatorial Theory B 40 (1986) 330–343.

    Article  MATH  Google Scholar 

  24. M. Grötschel, L. Lovász and A. Schrijver,Geometric Algorithms and Combinatorial Optimization (Springer, Berlin, 1988).

    MATH  Google Scholar 

  25. J. Håstad, Clique is hard to approximate withinn 1–ε, in:Proceedings of the 37th Symposium on Foundations of Computer Science (1996) 627–636.

  26. J. Håstad, Some optimal inapproximability results, in:Proceedings of the 29th ACM Symposium on Theory of Computing (1997) pp. 1–10.

  27. D.S. Hochbaum, Approximation algorithms for the set covering and vertex cover problems,SIAM Journal on Computing 11 (1982) 555–556.

    Article  MATH  MathSciNet  Google Scholar 

  28. R. Horn and C. Johnson,Matrix Analysis (Cambridge University Press, Cambridge, 1985).

    MATH  Google Scholar 

  29. F. Juhász, The asymptotic behavior of Lovász’v function for random graphs,Combinatorica (1982) 153–155.

  30. D. Karger, R. Motwani and M. Sudan, Approximate graph coloring by semidefinite programming, in:Proceedings 35th of the Symposium on Foundations of Computer Science (1994) 2–13.

  31. H. Karloff, How good is the Goemans-Williamson MAX CUT algorithm, in:Proceedings of the 28th ACM Symposium on Theory of Computing (1996) 427–434.

  32. J. Kleinberg and M.X. Goemans, The Lovász theta function and a semidefinite programming relaxation of vertex cover,SIAM Journal on Discrete Mathematics, to appear.

  33. D. Knuth, The sandwich theorem,Electronic Journal of Combinatorics 1 (1994); http://ejc.math.gatech.edu:8000/Journal/journalhome.html

  34. M. Kojima, Interior-point methods for semidefinite programming,Mathematical Programming (1997) (this volume).

  35. J. Lagergren and A. Russell, Vertex cover eludes the Schrijver function, Manuscript, 1997.

  36. P. Lancaster and M. Tismenetsky,The Theory of Matrices (Academic Press, Orlando, FL, 1985).

    MATH  Google Scholar 

  37. M. Laurent, S. Poljak and F. Rendl, Connections between the semidefinite relaxations of the max-cut and stable set problems,Mathematical Programming 77 (1997) 225–246.

    Article  MathSciNet  Google Scholar 

  38. F. Leighton and S. Rao, An approximate max-flow min-cut theorem for uniform multicommodity flow problems with applications to approximation algorithms, in:Proceedings of the 29th Symposium on Foundations of Computer Science (1988) 422–431.

  39. A. Lewis and M. Overton, Eigenvalue optimization,Acta Numerica 5 (1996) 149–190.

    Article  MathSciNet  Google Scholar 

  40. N. Linial, E. London and Y. Rabinovich, The geometry of graphs and some of its algorithmic applications,Combinatorica 15 (1995) 215–246.

    Article  MATH  MathSciNet  Google Scholar 

  41. L. Lovász, On the Shannon capacity of a graph,IEEE Transactions on Information Theory 25 (1979) 1–7.

    Article  MATH  Google Scholar 

  42. L. Lovász, Perfect graphs, in: L. Beineke and R. Wilson, eds.,Selected Topics in Graph Theory, Vol. 2 (Academic Press, New York, 1983) 55–87.

    Google Scholar 

  43. L. Lovász, Combinatorial optimization: Some problems and trends, Technical Report 92–53, DIMACS, 1992.

  44. L. Lovász and A. Schrijver, Matrix cones, projection representations, and stable set polyhedra, in:Polyhedral Combinatorics, DIMACS Series in Discrete Mathematics and Theoretical Computer Science, Vol. 1 (AMS, Providence, RI, 1989) 1–17.

    Google Scholar 

  45. L. Lovász and A. Schrijver, Cones of matrices and setfunctions, and 0–1 optimization,SIAM Journal on Optimization 1 (1991) 166–190.

    Article  MATH  MathSciNet  Google Scholar 

  46. F. MacWilliams and N. Sloane,The Theory of Error Correcting Codes (North-Holland, Amsterdam, 1977).

    MATH  Google Scholar 

  47. S. Mahajan and H. Ramesh, Derandomizing semidefinite programming based approximation algorithms, in:Proceedings 36th Symposium on Foundations of Computer Science (1995) 162–169.

  48. G. Malajovich, An effective version of Kronecker’s theorem on simultaneous Diophantine approximation, Technical Report, City University of Hong Kong, 1996.

  49. R. McEliece, The bounds of Delsarte and Lovász, and their applications to coding theory, in: G. Longo, ed.,Algebraic Coding Theory and Applications (Springer, Berlin, 1979) 107–178.

    Google Scholar 

  50. R. McEliece, E. Rodemich, H. Rumsey and L. Welch, New upper bounds on the rate of a code via the Delsarte-McWilliams inequalities,IEEE Transactions on Information Theory 23 (1977) 157.

    Article  MATH  MathSciNet  Google Scholar 

  51. B. Mohar and S. Poljak, Eigenvalue methods in combinatorial optimization, in: R. Brualdi, S. Friedland and V. Klee, eds.,Combinatorial and Graph-Theoretic Problems in Linear Algebra, The IMA Volumes in Mathematics and its Applications, Vol. 50 (Springer, Berlin, 1993) 107–151.

    Google Scholar 

  52. Y. Nesterov, Quality of semidefinite relaxation for nonconvex quadratic optimization, Manuscript, 1997.

  53. Y. Nesterov and A. Nemirovskii,Interior Point Polynomial Methods in Convex Programming (SIAM, Philadelphia, PA, 1994).

    Google Scholar 

  54. G. Pataki, On cone-LP’s and semi-definite programs: Facial structure, basic solutions, and the simplex method, GSIA Working Paper WP 1995–03, Carnegie-Mellon University, 1995.

  55. S. Poljak, Polyhedral and eigenvalue approximations of the max-cut problem, in: G. Halász et al., eds.,Sets, Graphs, and Numbers, Colloquia Mathematica Societatis János Bolyai, Vol. 60 (North-Holland, Amsterdam, 1992) 569–581.

    Google Scholar 

  56. S. Poljak and F. Rendl, Nonpolyhedral relaxations of graph-bisection problems,SIAM Journal on Optimization 5 (1995) 467–487.

    Article  MATH  MathSciNet  Google Scholar 

  57. S. Poljak and Z. Tuza, Maximum cuts and largest bipartite subgraphs, in: W. Cook, L. Lovász and P. Seymour, eds.,Combinatorial Optimization, DIMACS Series in Discrete Mathematics and Theoretical Computer Science (AMS, Providence, RI, 1995).

    Google Scholar 

  58. F. Rendl, Semidefinite programming and combinatorial optimization, Notes for a lecture series given at the University of Pisa, 1997.

  59. A. Schrijver, A comparison of the Delsarte and Lovász bounds,IEEE Transactions on Information Theory 25 (1979) 425–429.

    Article  MATH  MathSciNet  Google Scholar 

  60. C. Shannon, The zero error capacity of a channel,IRE Transactions on Information Theory 2 (1956) 8–19.

    Article  MathSciNet  Google Scholar 

  61. H. Sherali and W. Adams, A hierarchy of relaxations between the continuous and convex hull representations for zero-one programming problems,SIAM Journal on Discrete Mathematics 3 (1990) 411–430.

    Article  MATH  MathSciNet  Google Scholar 

  62. A. Sinclair and M. Jerrum, Approximate counting, uniform generation and rapidly mixing markov chains,Information and Computation 82 (1989) 93–133.

    Article  MATH  MathSciNet  Google Scholar 

  63. M. Szegedy, A note on the theta number of Lovász and the generalized Delsarte bound, in:Proceedings of the 35th Symposium on Foundations of Computer Science (1994) 36–39.

  64. P. Tiwari, A problem that is easier to solve on the unit-cost algebraic RAM,Journal of Complexity 8 (1992) 393–397.

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  66. M. Yannakakis, Expressing combinatorial optimization problems by linear programs, in:Proceedings of the 29th Symposium on Foundations of Computer Science (1988) 223–228.

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, MIT, Room 2-392, 02139, Cambridge, MA, USA

    Michel X. Goemans

Authors
  1. Michel X. Goemans
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Part of this work is supported by NSF contract 9623859-CCR, a Sloan Foundation Fellowship, and ARPA Contract N00014-95-1-1246.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Goemans, M.X. Semidefinite programming in combinatorial optimization. Mathematical Programming 79, 143–161 (1997). https://doi.org/10.1007/BF02614315

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02614315

Keywords

Search

Navigation