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Semidefinite Programs and Combinatorial Optimization

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Recent Advances in Algorithms and Combinatorics

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.

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References

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

    Article  MATH  MathSciNet  Google Scholar 

  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. 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. N. Alon and J.H. Spencer: The Probabilistic Method, Wiley, New York, 1992.

    MATH  Google Scholar 

  6. N. Alon, The Shannon capacity of a union, Combinatorica 18 (1998), 301–310.

    Article  MATH  MathSciNet  Google Scholar 

  7. N. Alon: Explicit Ramsey graphs and orthonormal labelings, The Electronic Journal of Combinatorics 1 (1994), 8pp.

    Google Scholar 

  8. N. Alon and N. Kahale: Approximating the independence number via the ϑ-function, Math. Programming 80 (1998), Ser. A, 253–264.

    MATH  MathSciNet  Google Scholar 

  9. E. Andre’ev, On convex polyhedra in Lobachevsky spaces, Mat. Sbornik, Nov. Ser. 81 (1970), 445–478.

    Google Scholar 

  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. 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. 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  MATH  MathSciNet  Google Scholar 

  13. A.I. Barvinok: Feasibility testing for systems of real quadratic equations, Discrete and Comp. Geometry 10 (1993), 1–13.

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  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.

    Article  MATH  MathSciNet  Google Scholar 

  16. J. Beck: Roth’s estimate on the discrepancy of integer sequences is nearly sharp, Combinatorica 1 (1981) 327–335.

    Article  MATH  MathSciNet  Google Scholar 

  17. J. Beck and W. Chen: Irregularities of Distribution, Cambridge Univ. Press (1987).

    Book  MATH  Google Scholar 

  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. M. Bellare, O. Goldreich, M. Sudan: Free bits, PCPs and non-approximability — towards tight results, Proc. 36th FOCS (1996), 422–431.

    Google Scholar 

  20. A. Blum and D. Karger: An O(n 3/14)-coloring for 3-colorable graphs, Inform. Process. Lett. 61 (1997), 49–53.

    Article  MathSciNet  Google Scholar 

  21. R. Boppana and M. Haldórsson: Approximating maximum independent sets by excluding subgraps, BIT 32 (1992), 180–196.

    Article  MATH  MathSciNet  Google Scholar 

  22. M. Boulala and J.-P. Uhry: Polytope des indépendants d’un graphe série-parallèle, Discrete Math. 27 (1979), 225–243.

    Article  MATH  MathSciNet  Google Scholar 

  23. V. Chvátal: On certain polytopes associated with graphs, J. of Combinatorial Theory (B) 18 (1975), 138–154.

    Article  MATH  Google Scholar 

  24. Y. Colin de Verdière, Sur la multiplicité de la première valeur propre non nulle du laplacien, Comment. Math. Helv. 61 (1986), 254–270.

    Article  MATH  MathSciNet  Google Scholar 

  25. Y. Colin de Verdière, Sur un novel invariant des graphes at un critère de planarité, J. Combin. Theory B 50 (1990) 11–21.

    Article  MATH  Google Scholar 

  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.

    Chapter  Google Scholar 

  27. M. Deza and M. Laurent: Geometry of Cuts and Metrics, Springer Verlag, 1997.

    MATH  Google Scholar 

  28. C. Delorme and S. Poljak: Combinatorial properties and the complexity of max-cut approximations, Europ. J. Combin. 14 (1993), 313–333.

    Article  MATH  MathSciNet  Google Scholar 

  29. C. Delorme and S. Poljak: Laplacian eigenvalues and the maximum cut problem, Math. Programming 62 (1993)

    Google Scholar 

  30. P. Erdős: Gráfok páros körüljárasú részgráfjairól (On bipartite subgraphs of graphs, in Hungarian), Mat. Lapok 18 (1967), 283–288.

    MathSciNet  Google Scholar 

  31. P. Erdős, F. Harary and W.T. Tutte, On the dimension of a graph Mathematika 12 (1965), 118–122.

    Article  MathSciNet  Google Scholar 

  32. U. Feige: Randomized graph products, chromatic numbers, and the Lovász ϑ-function, Combinatorica 17 (1997), 79–90.

    Article  MATH  MathSciNet  Google Scholar 

  33. U. Feige: Approximating the Bandwidth via Volume Respecting Embeddings, Tech. Report CS98–03, Weizmann Institute (1998).

    Google Scholar 

  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.

    Chapter  Google Scholar 

  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. S. Friedland and R. Loewy, Subspaces of symmetric matrices containing matrices with multiple first eigenvalue, Pacific J. Math. 62 (1976), 389–399.

    Article  MATH  MathSciNet  Google Scholar 

  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. 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.

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  40. 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 

  41. M. Grötschel, L. Lovász and A. Schrijver: Polynomial algorithms for perfect graphs, Annals of Discrete Math. 21 (1984), 325–256.

    Google Scholar 

  42. M. Grötschel, L. Lovász and A. Schrijver: Relaxations of vertex packing, J. Combin. Theory B 40 (1986), 330–343.

    Article  MATH  Google Scholar 

  43. M. Grötschel, L. Lovász and A. Schrijver: Geometric Algorithms and Combinatorial Optimization, Springer, Heidelberg, 1988.

    Book  MATH  Google Scholar 

  44. W. Haemers: On some problems of Lovász concerning the Shannon capacity of a graph, IEEE Trans. Inform. Theory 25 (1979), 231–232.

    Article  MATH  MathSciNet  Google Scholar 

  45. J. Håstad: Some optimal in-approximability results, Proc. 29th ACM Symp. on Theory of Comp., 1997, 1–10.

    Google Scholar 

  46. J. Håstad: Clique is hard to approximate within a factor of n 1−ε , Acta Math. 182 (1999), 105–142.

    Article  MATH  MathSciNet  Google Scholar 

  47. H. van der Holst, A short proof of the planarity characterization of Colin de Verdière, Preprint, CWI Amsterdam, 1994.

    Google Scholar 

  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.

    Article  Google Scholar 

  49. F. Juhász: The asymptotic behaviour of Lovász’ ϑ function for random graphs, Combinatorica 2 (1982) 153–155.

    Article  MATH  MathSciNet  Google Scholar 

  50. N. Kahale: A semidefinite bound for mixing rates of Markov chains, DIMACS Tech. Report No. 95–41.

    Google Scholar 

  51. D. Karger, R. Motwani, M. Sudan: Approximate graph coloring by semidefinite programming, Proc. 35th FOCS (1994), 2–13

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  53. H. Karloff: How good is the Goemans-Williamson MAX CUT algorithm? SIAM J. Comput. 29 (1999), 336–350.

    Article  MATH  MathSciNet  Google Scholar 

  54. 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. B. S. Kashin and S. V. Konyagin: On systems of vectors in Hilbert spaces, Trudy Mat. Inst. V.A.Steklova 157 (1981), 64–67.

    MATH  MathSciNet  Google Scholar 

  56. B. S. Kashin and S. V. Konyagin: English translation: Proc. of the Steklov Inst. of Math. (AMS 1983), 67–70.

    Google Scholar 

  57. V. S. Konyagin, Systems of vectors in Euclidean space and an extremal problem for polynomials, Mat. Zametky 29 (1981), 63–74

    MATH  MathSciNet  Google Scholar 

  58. 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.

    Article  MATH  MathSciNet  Google Scholar 

  59. A. Kotlov, L. Lovász, S. Vempala, The Colin de Verdière number and sphere representations of a graph, Combinatorica 17 (1997) 483–521.

    Article  MathSciNet  Google Scholar 

  60. D. E. Knuth: The sandwich theorem, The Electronic Journal of Combinatorics 1 (1994) 48 pp.

    Google Scholar 

  61. 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. 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.

    Article  MATH  MathSciNet  Google Scholar 

  63. N. Linial, L. Lovász, A. Wigderson: Rubber bands, convex embeddings, and graph connectivity, Combinatorica 8 (1988), 91–102.

    Article  MATH  MathSciNet  Google Scholar 

  64. L. Lipták, L. Lovász: Facets with fixed defect of the stable set polytope, Math. Programming, Series A 88 (2000), 33–44.

    Article  MATH  Google Scholar 

  65. L. Lovász: Normal hypergraphs and the perfect graph conjecture, Discrete Math. 2 (1972), 253–267.

    Article  MATH  MathSciNet  Google Scholar 

  66. L. Lovász: Some finite basis theorems in graph theory, in: Combinatorics, Coll. Math. Soc. J. Bolyai 18 (1978), 717–729.

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  68. 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. L. Lovász: Singular spaces of matrices and their applications in combinatorics, Bol. Soc. Braz. Mat. 20 (1989), 87–99.

    Article  MATH  Google Scholar 

  70. L. Lovász: Stable sets and polynomials, Discrete Math. 124 (1994), 137–153.

    Article  MATH  MathSciNet  Google Scholar 

  71. L. Lovász: Integer sequences and semidefinite programming Publ. Math. Debrecen 56 (2000) 475–479.

    MATH  MathSciNet  Google Scholar 

  72. L. Lovász, M. Saks and A. Schrijver: Orthogonal representations and connectivity of graphs, Linear Alg. Appl. 114/115 (1989), 439–454.

    Google Scholar 

  73. 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.

    Article  MATH  MathSciNet  Google Scholar 

  74. L. Lovász and A. Schrijver: Cones of matrices and set-functions, and 0–1 optimization, SIAM J. on Optimization 1 (1990), 166–190.

    Article  Google Scholar 

  75. 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. L. Lovász and K. Vesztergombi: Geometric representations of graphs, in: Paul Erdős and his Mathematics

    Google Scholar 

  77. J. Matoušek and J. Spencer, Discrepancy in arithmetic progressions, J. Amer. Math. Soc. 9 (1996) 195–204.

    Article  MATH  MathSciNet  Google Scholar 

  78. B. Mohar and S. Poljak: Eigenvalues and the max-cut problem, Czechoslovak Mathematical Journal 40 (1990), 343–352.

    MathSciNet  Google Scholar 

  79. Yu. E. Nesterov and A. Nemirovsky: Interior-point polynomial methods in convex programming, Studies in Appl. Math. 13, SIAM, Philadelphia, 1994.

    Google Scholar 

  80. M. L. Overton: On minimizing the maximum eigenvalue of a symmetric matrix, SIAM J. on Matrix Analysis and Appl. 9 (1988), 256–268.

    Article  MATH  MathSciNet  Google Scholar 

  81. 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.

    Article  MATH  MathSciNet  Google Scholar 

  82. M. Padberg: Linear optimization and extensions. Second, revised and expanded edition, Algorithms and Combinatorics 12, Springer-Verlag, Berlin, 1999.

    Book  MATH  Google Scholar 

  83. 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.

    Article  MATH  MathSciNet  Google Scholar 

  84. S. Poljak and F. Rendl: Nonpolyhedral relaxations of graph-bisection problems, DIMACS Tech. Report 92–55 (1992).

    Google Scholar 

  85. L. Porkoláb and L. Khachiyan: On the complexity of semidefinite programs, J. Global Optim. 10 (1997), 351–365.

    Article  MATH  MathSciNet  Google Scholar 

  86. 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. A. Recski: Matroid Theory and its Applications in Electric Network Theory and Statics, Akadémiai Kiadó-Springer-Verlag (1989).

    Book  Google Scholar 

  88. K.F. Roth: Remark concerning integer sequences, Acta Arith. 35, 257–260.

    Google Scholar 

  89. O. Schramm: How to cage an egg, Invent. Math. 107 (1992), 543–560.

    Article  MATH  MathSciNet  Google Scholar 

  90. 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.

    Article  MATH  MathSciNet  Google Scholar 

  91. G. Strang: Linear algebra and its applications, Second edition, Academic Press, New York-London, 1980.

    Google Scholar 

  92. M. Szegedy: A note on the θ number of Lovász and the generalized Delsarte bound, Proc. 35th FOCS (1994), 36–39.

    Google Scholar 

  93. E. Tardos: The gap between monotone and non-monotone circuit complexity is exponential, Combinatorica 8 (1988), 141–142.

    Article  MATH  MathSciNet  Google Scholar 

  94. W. Thurston: The Geometry and Topology of Three-manifolds, Princeton Lecture Notes, Chapter 13, Princeton, 1985.

    Google Scholar 

  95. W.T. Tutte: How to draw a graph, Proc. London Math. Soc. 13 (1963), 743–768.

    Article  MATH  MathSciNet  Google Scholar 

  96. 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. L. Vandeberghe and S. Boyd: Semidefinite programming. SIAM Rev. 38 (1996), no. 1, 49–95.

    Article  MathSciNet  Google Scholar 

  98. R.J. Vanderbei and B. Yang: The simplest semidefinite programs are trivial, Math. of Oper. Res. 20 (1995), no. 3, 590–596.

    Article  MATH  MathSciNet  Google Scholar 

  99. H. Wolkowitz: Some applications of optimization in matrix theory, Linear Algebra and its Applications 40 (1981), 101–118.

    Article  MathSciNet  Google Scholar 

  100. H. Wolkowicz: Explicit solutions for interval semidefinite linear programs, Linear Algebra Appl. 236 (1996), 95–104.

    Article  MATH  MathSciNet  Google Scholar 

  101. 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.

    Book  Google Scholar 

  102. 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 

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  1. L. Lovász
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Editors and Affiliations

  1. Equipe Combinatoire, CNRS, Paris, France

    Bruce A. Reed

  2. McGill University, Montreal, Canada

    Bruce A. Reed

  3. Departamento de Computacao—LIA, Universidade Federal do Ceara, Campus do Pici—Bloco 910, CEP 60455-760, Fortaleza, CE, Brasil

    Cláudia L. Sales

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Lovász, L. (2003). Semidefinite Programs and Combinatorial Optimization. In: Reed, B.A., Sales, C.L. (eds) Recent Advances in Algorithms and Combinatorics. CMS Books in Mathematics / Ouvrages de mathématiques de la SMC. Springer, New York, NY. https://doi.org/10.1007/0-387-22444-0_6

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