Semidefinite Programs and Combinatorial Optimization

  • L. Lovász

Abstract

Linear programming has been one of the most fundamental and successful tools in optimization and discrete mathematics. Its applications include exact and approximation algorithms, as well as structural results and estimates. The key point is that linear programs are very efficiently solvable, and have a powerful duality theory.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    F. Alizadeh: Combinatorial optimization with semi-definite matrices, in: Integer Programming and Combinatorial Optimization (Proceedings of IPCO ’92), (eds. E. Balas, G. Cornuéjols and R. Kannan), Carnegie Mellon University Printing (1992), 385–405.Google Scholar
  2. [2]
    F. Alizadeh, Interior point methods in semidefinite programming with applications to combinatorial optimization, SIAM J. Optim. 5 (1995), 13–51.CrossRefMATHMathSciNetGoogle Scholar
  3. [3]
    F. Alizadeh, J.-P. Haeberly, and M. Overton: Complementarity and nonde-generacy in semidefinite programming, in: Semidefinite Programming, Math. Programming Ser. B, 77 (1997), 111–128.Google Scholar
  4. [4]
    N. Alon, R. A. Duke, H. Lefmann, V. Rödl and R. Yuster: The algorithmic aspects of the Regularity Lemma, Proc. 33rd Annual Symp. on Found, of Computer Science, IEEE Computer Society Press (1992), 473–481.Google Scholar
  5. [5]
    N. Alon and J.H. Spencer: The Probabilistic Method, Wiley, New York, 1992.MATHGoogle Scholar
  6. [6]
    N. Alon, The Shannon capacity of a union, Combinatorica 18 (1998), 301–310.CrossRefMATHMathSciNetGoogle Scholar
  7. [7]
    N. Alon: Explicit Ramsey graphs and orthonormal labelings, The Electronic Journal of Combinatorics 1 (1994), 8pp.Google Scholar
  8. [8]
    N. Alon and N. Kahale: Approximating the independence number via the ϑ-function, Math. Programming 80 (1998), Ser. A, 253–264.MATHMathSciNetGoogle Scholar
  9. [9]
    E. Andre’ev, On convex polyhedra in Lobachevsky spaces, Mat. Sbornik, Nov. Ser. 81 (1970), 445–478.Google Scholar
  10. [10]
    S. Arora, C. Lund, R. Motwani, M. Sudan, M. Szegedy: Proof verification and hardness of approximation problems Proc. 33rd FOCS (1992), 14–23.Google Scholar
  11. [11]
    R. Bacher and Y. Colin de Verdière, Multiplicités de valeurs propres et transformations étoile-triangle des graphes, Bull. Soc. Math. France 123 (1995), 101–117.Google Scholar
  12. [12]
    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.CrossRefMATHMathSciNetGoogle Scholar
  13. [13]
    A.I. Barvinok: Feasibility testing for systems of real quadratic equations, Discrete and Comp. Geometry 10 (1993), 1–13.CrossRefMATHMathSciNetGoogle Scholar
  14. [14]
    A.I. Barvinok: Problems of distance geometry and convex properties of quadratic maps, Discrete and Comp. Geometry 13 (1995), 189–202.CrossRefMATHMathSciNetGoogle Scholar
  15. [15]
    A.I. Barvinok: A remark on the rank of positive semidefinite matrices subject to affine constraints, Discrete and Comp. Geometry 25 (2001), 23–31.CrossRefMATHMathSciNetGoogle Scholar
  16. [16]
    J. Beck: Roth’s estimate on the discrepancy of integer sequences is nearly sharp, Combinatorica 1 (1981) 327–335.CrossRefMATHMathSciNetGoogle Scholar
  17. [17]
    J. Beck and W. Chen: Irregularities of Distribution, Cambridge Univ. Press (1987).CrossRefMATHGoogle Scholar
  18. [18]
    J. Beck and V.T. Sós: Discrepancy Theory, Chapter 26 in: Handbook of Combinatorics (ed. R.L. Graham, M. Grötschel and L. Lovász), North-Holland, Amsterdam (1995).Google Scholar
  19. [19]
    M. Bellare, O. Goldreich, M. Sudan: Free bits, PCPs and non-approximability — towards tight results, Proc. 36th FOCS (1996), 422–431.Google Scholar
  20. [20]
    A. Blum and D. Karger: An O(n 3/14)-coloring for 3-colorable graphs, Inform. Process. Lett. 61 (1997), 49–53.CrossRefMathSciNetGoogle Scholar
  21. [21]
    R. Boppana and M. Haldórsson: Approximating maximum independent sets by excluding subgraps, BIT 32 (1992), 180–196.CrossRefMATHMathSciNetGoogle Scholar
  22. [22]
    M. Boulala and J.-P. Uhry: Polytope des indépendants d’un graphe série-parallèle, Discrete Math. 27 (1979), 225–243.CrossRefMATHMathSciNetGoogle Scholar
  23. [23]
    V. Chvátal: On certain polytopes associated with graphs, J. of Combinatorial Theory (B) 18 (1975), 138–154.CrossRefMATHGoogle Scholar
  24. [24]
    Y. Colin de Verdière, Sur la multiplicité de la première valeur propre non nulle du laplacien, Comment. Math. Helv. 61 (1986), 254–270.CrossRefMATHMathSciNetGoogle Scholar
  25. [25]
    Y. Colin de Verdière, Sur un novel invariant des graphes at un critère de planarité, J. Combin. Theory B 50 (1990) 11–21.CrossRefMATHGoogle Scholar
  26. [26]
    Y. Colin de Verdière, On a new graph invariant and a criterion for planarity, in: Graph Structure Theory (Robertson and P. D. Seymour, eds.), Contemporary Mathematics, Amer. Math. Soc, Providence, RI (1993), 137–147.CrossRefGoogle Scholar
  27. [27]
    M. Deza and M. Laurent: Geometry of Cuts and Metrics, Springer Verlag, 1997.MATHGoogle Scholar
  28. [28]
    C. Delorme and S. Poljak: Combinatorial properties and the complexity of max-cut approximations, Europ. J. Combin. 14 (1993), 313–333.CrossRefMATHMathSciNetGoogle Scholar
  29. [29]
    C. Delorme and S. Poljak: Laplacian eigenvalues and the maximum cut problem, Math. Programming 62 (1993)Google Scholar
  30. [30]
    P. Erdős: Gráfok páros körüljárasú részgráfjairól (On bipartite subgraphs of graphs, in Hungarian), Mat. Lapok 18 (1967), 283–288.MathSciNetGoogle Scholar
  31. [31]
    P. Erdős, F. Harary and W.T. Tutte, On the dimension of a graph Mathematika 12 (1965), 118–122.CrossRefMathSciNetGoogle Scholar
  32. [32]
    U. Feige: Randomized graph products, chromatic numbers, and the Lovász ϑ-function, Combinatorica 17 (1997), 79–90.CrossRefMATHMathSciNetGoogle Scholar
  33. [33]
    U. Feige: Approximating the Bandwidth via Volume Respecting Embeddings, Tech. Report CS98–03, Weizmann Institute (1998).Google Scholar
  34. [34]
    U. Feige and M. Goemans, Approximating the value of two-prover proof systems, with applications to MAX-2SAT and MAX-DICUT, in: Proc. 3rd Israel Symp. on Theory and Comp. Sys., Tel Aviv, Isr. (1995), 182–189.CrossRefGoogle Scholar
  35. [35]
    U. Feige and L. Lovász: Two-prover one-round proof systems: Their power and their problems. Proc. 24th ACM Symp. on Theory of Computing (1992), 733–744.Google Scholar
  36. [36]
    S. Friedland and R. Loewy, Subspaces of symmetric matrices containing matrices with multiple first eigenvalue, Pacific J. Math. 62 (1976), 389–399.CrossRefMATHMathSciNetGoogle Scholar
  37. [37]
    M. X. Goemans and D. P. Williamson: 878-Approximation algorithms for MAX CUT and MAX 2SAT, Proc. 26th ACM Symp. on Theory of Computing (1994), 422–431.Google Scholar
  38. [38]
    M. X. Goemans and D. P. Williamson: Improved approximation algorithms for maximum cut and satisfiablity problems using semidefinite programming, J. ACM 42 (1995), 1115–1145.CrossRefMATHMathSciNetGoogle Scholar
  39. [39]
    M. C. Golumbic: Algorithmic Graph Theory and Perfect Graphs, Academic Press, New York (1980).MATHGoogle Scholar
  40. [40]
    M. Grötschel, L. Lovász and A. Schrijver: The ellipsoid method and its consequences in combinatorial optimization, Combinatorica 1 (1981), 169–197.CrossRefMATHMathSciNetGoogle Scholar
  41. [41]
    M. Grötschel, L. Lovász and A. Schrijver: Polynomial algorithms for perfect graphs, Annals of Discrete Math. 21 (1984), 325–256.Google Scholar
  42. [42]
    M. Grötschel, L. Lovász and A. Schrijver: Relaxations of vertex packing, J. Combin. Theory B 40 (1986), 330–343.CrossRefMATHGoogle Scholar
  43. [43]
    M. Grötschel, L. Lovász and A. Schrijver: Geometric Algorithms and Combinatorial Optimization, Springer, Heidelberg, 1988.CrossRefMATHGoogle Scholar
  44. [44]
    W. Haemers: On some problems of Lovász concerning the Shannon capacity of a graph, IEEE Trans. Inform. Theory 25 (1979), 231–232.CrossRefMATHMathSciNetGoogle Scholar
  45. [45]
    J. Håstad: Some optimal in-approximability results, Proc. 29th ACM Symp. on Theory of Comp., 1997, 1–10.Google Scholar
  46. [46]
    J. Håstad: Clique is hard to approximate within a factor of n 1−ε , Acta Math. 182 (1999), 105–142.CrossRefMATHMathSciNetGoogle Scholar
  47. [47]
    H. van der Holst, A short proof of the planarity characterization of Colin de Verdière, Preprint, CWI Amsterdam, 1994.Google Scholar
  48. [48]
    H. van der Holst, L. Lovász and A. Schrijver: On the invariance of Colin de Verdière’s graph parameter under clique sums, Linear Algebra and its Applications, 226–228 (1995), 509–518.CrossRefGoogle Scholar
  49. [49]
    F. Juhász: The asymptotic behaviour of Lovász’ ϑ function for random graphs, Combinatorica 2 (1982) 153–155.CrossRefMATHMathSciNetGoogle Scholar
  50. [50]
    N. Kahale: A semidefinite bound for mixing rates of Markov chains, DIMACS Tech. Report No. 95–41.Google Scholar
  51. [51]
    D. Karger, R. Motwani, M. Sudan: Approximate graph coloring by semidefinite programming, Proc. 35th FOCS (1994), 2–13Google Scholar
  52. [51a]
    D. Karger, R. Motwani, M. Sudan: Approximate graph coloring by semidefinite programming, full version: J. ACM 45 (1998), 246–265.CrossRefMATHMathSciNetGoogle Scholar
  53. [52]
    H. Karloff: How good is the Goemans-Williamson MAX CUT algorithm? SIAM J. Comput. 29 (1999), 336–350.CrossRefMATHMathSciNetGoogle Scholar
  54. [53]
    H. Karloff and U. Zwick: A 7/8-approximation algorithm for MAX 3SAT? in: Proc. of the 38th Ann. IEEE Symp. in Found. of Comp. Sci. (1997), 406–415.Google Scholar
  55. [54]
    B. S. Kashin and S. V. Konyagin: On systems of vectors in Hilbert spaces, Trudy Mat. Inst. V.A.Steklova 157 (1981), 64–67.MATHMathSciNetGoogle Scholar
  56. [54a]
    B. S. Kashin and S. V. Konyagin: English translation: Proc. of the Steklov Inst. of Math. (AMS 1983), 67–70.Google Scholar
  57. [55]
    V. S. Konyagin, Systems of vectors in Euclidean space and an extremal problem for polynomials, Mat. Zametky 29 (1981), 63–74MATHMathSciNetGoogle Scholar
  58. [55a]
    V. S. Konyagin, Systems of vectors in Euclidean space and an extremal problem for polynomials. English translation: Math. Notes of the Academy USSR 29 (1981), 33–39.CrossRefMATHMathSciNetGoogle Scholar
  59. [56]
    A. Kotlov, L. Lovász, S. Vempala, The Colin de Verdière number and sphere representations of a graph, Combinatorica 17 (1997) 483–521.CrossRefMathSciNetGoogle Scholar
  60. [57]
    D. E. Knuth: The sandwich theorem, The Electronic Journal of Combinatorics 1 (1994) 48 pp.Google Scholar
  61. [58]
    P. Koebe: Kontaktprobleme der konformen Abbildung, Berichte uber die Verhandlungen d. Sächs. Akad. d. Wiss., Math.-Phys. Klasse, 88 (1936) 141–164.Google Scholar
  62. [59]
    M. Laurent and S. Poljak: On the facial structure of the set of correlation matrices, SIAM J. on Matrix Analysis and Applications 17 (1996), 530–547.CrossRefMATHMathSciNetGoogle Scholar
  63. [60]
    N. Linial, L. Lovász, A. Wigderson: Rubber bands, convex embeddings, and graph connectivity, Combinatorica 8 (1988), 91–102.CrossRefMATHMathSciNetGoogle Scholar
  64. [61]
    L. Lipták, L. Lovász: Facets with fixed defect of the stable set polytope, Math. Programming, Series A 88 (2000), 33–44.CrossRefMATHGoogle Scholar
  65. [62]
    L. Lovász: Normal hypergraphs and the perfect graph conjecture, Discrete Math. 2 (1972), 253–267.CrossRefMATHMathSciNetGoogle Scholar
  66. [63]
    L. Lovász: Some finite basis theorems in graph theory, in: Combinatorics, Coll. Math. Soc. J. Bolyai 18 (1978), 717–729.Google Scholar
  67. [64]
    L. Lovász: On the Shannon capacity of graphs, IEEE Trans. Inform. Theory 25 (1979), 1–7.CrossRefMATHMathSciNetGoogle Scholar
  68. [65]
    L. Lovász: Perfect graphs, in: More Selected Topics in Graph Theory (ed. L. W. Beineke, R. L. Wilson), Academic Press (1983), 55–67.Google Scholar
  69. [66]
    L. Lovász: Singular spaces of matrices and their applications in combinatorics, Bol. Soc. Braz. Mat. 20 (1989), 87–99.CrossRefMATHGoogle Scholar
  70. [67]
    L. Lovász: Stable sets and polynomials, Discrete Math. 124 (1994), 137–153.CrossRefMATHMathSciNetGoogle Scholar
  71. [68]
    L. Lovász: Integer sequences and semidefinite programming Publ. Math. Debrecen 56 (2000) 475–479.MATHMathSciNetGoogle Scholar
  72. [69]
    L. Lovász, M. Saks and A. Schrijver: Orthogonal representations and connectivity of graphs, Linear Alg. Appl. 114/115 (1989), 439–454.Google Scholar
  73. [70]
    L. Lovász, M. Saks and A. Schrijver: A correction: orthogonal representations and connectivity of graphs (with M. Saks and A. Schrijver) Linear Algebra Appl. 313 (2000) 101–105.CrossRefMATHMathSciNetGoogle Scholar
  74. [71]
    L. Lovász and A. Schrijver: Cones of matrices and set-functions, and 0–1 optimization, SIAM J. on Optimization 1 (1990), 166–190.CrossRefGoogle Scholar
  75. [72]
    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 I, Amer. Math. Soc, Providence (1990), 1–17.Google Scholar
  76. [73]
    L. Lovász and K. Vesztergombi: Geometric representations of graphs, in: Paul Erdős and his Mathematics Google Scholar
  77. [74]
    J. Matoušek and J. Spencer, Discrepancy in arithmetic progressions, J. Amer. Math. Soc. 9 (1996) 195–204.CrossRefMATHMathSciNetGoogle Scholar
  78. [75]
    B. Mohar and S. Poljak: Eigenvalues and the max-cut problem, Czechoslovak Mathematical Journal 40 (1990), 343–352.MathSciNetGoogle Scholar
  79. [76]
    Yu. E. Nesterov and A. Nemirovsky: Interior-point polynomial methods in convex programming, Studies in Appl. Math. 13, SIAM, Philadelphia, 1994.Google Scholar
  80. [77]
    M. L. Overton: On minimizing the maximum eigenvalue of a symmetric matrix, SIAM J. on Matrix Analysis and Appl. 9 (1988), 256–268.CrossRefMATHMathSciNetGoogle Scholar
  81. [78]
    M. L. Overton and R. Womersley: On the sum of the largest eigenvalues of a symmetric matrix, SIAM J. on Matrix Analysis and Appl. 13 (1992), 41–45.CrossRefMATHMathSciNetGoogle Scholar
  82. [79]
    M. Padberg: Linear optimization and extensions. Second, revised and expanded edition, Algorithms and Combinatorics 12, Springer-Verlag, Berlin, 1999.CrossRefMATHGoogle Scholar
  83. [80]
    G. Pataki: On the rank of extreme matrices in semidefinite programs and the multiplicity of optimal eigenvalues, Math. of Oper. Res. 23 (1998), 339–358.CrossRefMATHMathSciNetGoogle Scholar
  84. [81]
    S. Poljak and F. Rendl: Nonpolyhedral relaxations of graph-bisection problems, DIMACS Tech. Report 92–55 (1992).Google Scholar
  85. [82]
    L. Porkoláb and L. Khachiyan: On the complexity of semidefinite programs, J. Global Optim. 10 (1997), 351–365.CrossRefMATHMathSciNetGoogle Scholar
  86. [83]
    M. Ramana: An exact duality theory for semidefinite programming and its complexity implications, in: Semidefinite programming. Math. Programming Ser. B, 77 (1997), 129–162.Google Scholar
  87. [84]
    A. Recski: Matroid Theory and its Applications in Electric Network Theory and Statics, Akadémiai Kiadó-Springer-Verlag (1989).CrossRefGoogle Scholar
  88. [85]
    K.F. Roth: Remark concerning integer sequences, Acta Arith. 35, 257–260.Google Scholar
  89. [86]
    O. Schramm: How to cage an egg, Invent. Math. 107 (1992), 543–560.CrossRefMATHMathSciNetGoogle Scholar
  90. [87]
    H.D. Sherali and W.P. Adams (1990): A hierarchy of relaxations between the continuous and convex hull representations for zero-one programming problems, SIAM J. on Discrete Math. bf 3, 411–430.CrossRefMATHMathSciNetGoogle Scholar
  91. [88]
    G. Strang: Linear algebra and its applications, Second edition, Academic Press, New York-London, 1980.Google Scholar
  92. [89]
    M. Szegedy: A note on the θ number of Lovász and the generalized Delsarte bound, Proc. 35th FOCS (1994), 36–39.Google Scholar
  93. [90]
    E. Tardos: The gap between monotone and non-monotone circuit complexity is exponential, Combinatorica 8 (1988), 141–142.CrossRefMATHMathSciNetGoogle Scholar
  94. [91]
    W. Thurston: The Geometry and Topology of Three-manifolds, Princeton Lecture Notes, Chapter 13, Princeton, 1985.Google Scholar
  95. [92]
    W.T. Tutte: How to draw a graph, Proc. London Math. Soc. 13 (1963), 743–768.CrossRefMATHMathSciNetGoogle Scholar
  96. [93]
    L. Vandeberghe and S. Boyd: Semidefinite programming, in: Math. Programming: State of the Art (ed. J. R. Birge and K. G. Murty), Univ. of Michigan, 1994.Google Scholar
  97. [94]
    L. Vandeberghe and S. Boyd: Semidefinite programming. SIAM Rev. 38 (1996), no. 1, 49–95.CrossRefMathSciNetGoogle Scholar
  98. [95]
    R.J. Vanderbei and B. Yang: The simplest semidefinite programs are trivial, Math. of Oper. Res. 20 (1995), no. 3, 590–596.CrossRefMATHMathSciNetGoogle Scholar
  99. [96]
    H. Wolkowitz: Some applications of optimization in matrix theory, Linear Algebra and its Applications 40 (1981), 101–118.CrossRefMathSciNetGoogle Scholar
  100. [97]
    H. Wolkowicz: Explicit solutions for interval semidefinite linear programs, Linear Algebra Appl. 236 (1996), 95–104.CrossRefMATHMathSciNetGoogle Scholar
  101. [98]
    H. Wolkowicz, R. Saigal and L. Vandenberghe: Handbook of semidefinite programming. Theory, algorithms, and applications. Int. Ser. Oper. Res. & Man. Sci., 27 (2000) Kluwer Academic Publishers, Boston, MA.CrossRefGoogle Scholar
  102. [99]
    U. Zwick: Outward rotations: a tool for rounding solutions of semidefinite programming relaxations, with applications to MAX CUT and other problems, Proc. 31th STOC (1999), 679–687.Google Scholar

Copyright information

© Springer-Verlag New York, Inc. 2003

Authors and Affiliations

  • L. Lovász

There are no affiliations available

Personalised recommendations