Semidefinite Programs and Combinatorial Optimization

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.

[1]  Charles Delorme,et al.  Combinatorial Properties and the Complexity of a Max-cut Approximation , 1993, Eur. J. Comb..

[2]  N. Alon,et al.  The Algorithmic Aspects of the Regularity Lemma (Extended Abstract) , 1992, FOCS 1992.

[3]  László Lovász,et al.  Singular spaces of matrices and their application in combinatorics , 1989 .

[4]  H. Wolkowicz,et al.  Some applications of optimization in matrix theory , 1981 .

[5]  László Lovász.  Integer Sequences and Semidefinite Programming , 2000 .

[6]  K. F. Roth.  On irregularities of distribution , 1954 .

[7]  David P. Williamson,et al.  Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming , 1995, JACM.

[8]  Michel Deza,et al.  Geometry of cuts and metrics , 2009, Algorithms and combinatorics.

[9]  R. Saigal,et al.  Handbook of semidefinite programming : theory, algorithms, and applications , 2000 .

[10]  Hein Vanderholst,et al.  A Short Proof of the Planarity Characterization of Colin de Verdière , 1995, J. Comb. Theory, Ser. B.

[11]  Johan Håstad,et al.  Some optimal inapproximability results , 1997, STOC '97.

[12]  William W. L. Chen.  On irregularities of distribution. , 1980 .

[13]  Jiří Matoušek,et al.  Discrepancy in arithmetic progressions , 1996 .

[14]  David Kargery.  O(n 3=14 )-coloring for 3-colorable Graphs , 1996 .

[15]  Y. C. Verdière,et al.  Sur la multiplicité de la première valeur propre non nulle du Laplacien , 1986 .

[16]  R. Bacher,et al.  Multiplicités des valeurs propres et transformations étoile-triangle des graphes , 1995 .

[17]  L. Lovász,et al.  Orthogonal representations and connectivity of graphs , 1989 .

[18]  József Beck,et al.  Irregularities of distribution: Index of theorems and corollaries , 1987 .

[19]  József Beck,et al.  Balanced two-colorings of finite sets in the square I , 1981, Comb..

[20]  József Beck,et al.  Roth’s estimate of the discrepancy of integer sequences is nearly sharp , 1981, Comb..

[21]  J. Beck,et al.  Irregularities of distribution , 1987 .

[22]  László Lovász,et al.  On the Shannon capacity of a graph , 1979, IEEE Trans. Inf. Theory.

[23]  A. Recski.  Matroid theory and its applications in electric network theory and in statics , 1989 .

[24]  Ferenc Juhász,et al.  The asymptotic behaviour of lovász’ ϑ function for random graphs , 1982, Comb..

[25]  Alexander I. Barvinok,et al.  A Remark on the Rank of Positive Semidefinite Matrices Subject to Affine Constraints , 2001, Discret. Comput. Geom..

[26]  Carsten Lund,et al.  Proof verification and hardness of approximation problems , 1992, Proceedings., 33rd Annual Symposium on Foundations of Computer Science.

[27]  Alexander I. Barvinok,et al.  Feasibility testing for systems of real quadratic equations , 1992, STOC '92.

[28]  Uriel Feige,et al.  Approximating the Bandwidth via Volume Respecting Embeddings , 2000, J. Comput. Syst. Sci..

[29]  Alexander I. Barvinok,et al.  Problems of distance geometry and convex properties of quadratic maps , 1995, Discret. Comput. Geom..

[30]  Yves Colin de Verdière,et al.  On a new graph invariant and a criterion for planarity , 1991, Graph Structure Theory.

[31]  László Lovász,et al.  Rubber bands, convex embeddings and graph connectivity , 1988, Comb..

[32]  Mouloud Boulala,et al.  Polytope des independants d'un graphe serie-parallele , 1979, Discret. Math..

[33]  Yves Colin de Verdière,et al.  Sur un nouvel invariant des graphes et un critère de planarité , 1990, J. Comb. Theory, Ser. B.

[34]  Alexander Schrijver,et al.  Cones of Matrices and Set-Functions and 0-1 Optimization , 1991, SIAM J. Optim..

[35]  Mario Szegedy,et al.  A note on the /spl theta/ number of Lovasz and the generalized Delsarte bound , 1994, Proceedings 35th Annual Symposium on Foundations of Computer Science.

[36]  Farid Alizadeh,et al.  Interior Point Methods in Semidefinite Programming with Applications to Combinatorial Optimization , 1995, SIAM J. Optim..

[37]  Ravi B. Boppana,et al.  Approximating maximum independent sets by excluding subgraphs , 1992, BIT Comput. Sci. Sect..

[38]  W. Thurston.  The geometry and topology of three-manifolds , 1979 .

[39]  J. G. Pierce,et al.  Geometric Algorithms and Combinatorial Optimization , 2016 .

[40]  S. Konyagin.  Systems of vectors in Euclidean space and an extremal problem for polynomials , 1981 .

[41]  Willem H. Haemers,et al.  On Some Problems of Lovász Concerning the Shannon Capacity of a Graph , 1979, IEEE Trans. Inf. Theory.

[42]  Shmuel Friedland,et al.  Subspaces of symmetric matrices containing matrices with a multiple first eigenvalue. , 1976 .

[43]  H. Wolkowicz,et al.  EXPLICIT SOLUTIONS FOR INTERVAL SEMIDEFINITE LINEAR PROGRAMS , 1996 .

[44]  Gábor Pataki,et al.  On the Rank of Extreme Matrices in Semidefinite Programs and the Multiplicity of Optimal Eigenvalues , 1998, Math. Oper. Res..

[45]  Geoffrey Exoo,et al.  A Lower Bound for Schur Numbers and Multicolor Ramsey Numbers , 1994, Electron. J. Comb..

[46]  J. Håstad.  Clique is hard to approximate withinn1−ε , 1999 .

[47]  M. Overton.  On minimizing the maximum eigenvalue of a symmetric matrix , 1988 .

[48]  Warren P. Adams,et al.  A hierarchy of relaxation between the continuous and convex hull representations , 1990 .

[49]  K. F. Roth.  Remark concerning integer sequences , 1964 .

[50]  Leonid Khachiyan,et al.  On the Complexity of Semidefinite Programs , 1997, J. Glob. Optim..

[51]  Noga Alon,et al.  The Probabilistic Method , 2015, Fundamentals of Ramsey Theory.

[52]  Howard J. Karloff,et al.  How good is the Goemans-Williamson MAX CUT algorithm? , 1996, STOC '96.

[53]  László Lovász,et al.  Two-prover one-round proof systems: their power and their problems (extended abstract) , 1992, STOC '92.

[54]  Charles Delorme,et al.  Laplacian eigenvalues and the maximum cut problem , 1993, Math. Program..

[55]  Martin Grötschel,et al.  The ellipsoid method and its consequences in combinatorial optimization , 1981, Comb..

[56]  L. Lovász,et al.  Polynomial Algorithms for Perfect Graphs , 1984 .

[57]  Santosh S. Vempala,et al.  The Colin de Verdière number and sphere representations of a graph , 1997, Comb..

[58]  Alexander Schrijver,et al.  A correction: orthogonal representations and connectivity of graphs , 2000 .

[59]  Monique Laurent,et al.  On the Facial Structure of the Set of Correlation Matrices , 1996, SIAM J. Matrix Anal. Appl..

[60]  Uri Zwick,et al.  A 7/8-approximation algorithm for MAX 3SAT? , 1997, Proceedings 38th Annual Symposium on Foundations of Computer Science.

[61]  Uriel Feige,et al.  Approximating the value of two power proof systems, with applications to MAX 2SAT and MAX DICUT , 1995, Proceedings Third Israel Symposium on the Theory of Computing and Systems.

[62]  Alexander Schrijver,et al.  Matrix Cones, Projection Representations, and Stable Set Polyhedra , 1990, Polyhedral Combinatorics.

[63]  Michael L. Overton,et al.  Complementarity and nondegeneracy in semidefinite programming , 1997, Math. Program..

[64]  Henry Wolkowicz,et al.  Handbook of Semidefinite Programming , 2000 .

[65]  Franz Rendl,et al.  Nonpolyhedral Relaxations of Graph-Bisection Problems , 1995, SIAM J. Optim..

[66]  R. Möhring.  Algorithmic graph theory and perfect graphs , 1986 .

[67]  Egon Balas,et al.  A lift-and-project cutting plane algorithm for mixed 0–1 programs , 1993, Math. Program..

[68]  Mihir Bellare,et al.  Free bits, PCPs and non-approximability-towards tight results , 1995, Proceedings of IEEE 36th Annual Foundations of Computer Science.

[69]  Noga Alon,et al.  Approximating the independence number via theϑ-function , 1998, Math. Program..

[70]  Donald E. Knuth.  The Sandwich Theorem , 1994, Electron. J. Comb..

[71]  Motakuri V. Ramana,et al.  An exact duality theory for semidefinite programming and its complexity implications , 1997, Math. Program..

[72]  Noga Alon,et al.  The Shannon Capacity of a Union , 1998, Comb..

[73]  Michael L. Overton,et al.  On the Sum of the Largest Eigenvalues of a Symmetric Matrix , 1992, SIAM J. Matrix Anal. Appl..

[74]  Uriel Feige,et al.  Randomized graph products, chromatic numbers, and the Lovász ϑ-function , 1997, Comb..

[75]  W. T. Tutte,et al.  ON THE DIMENSION OF A GRAPH , 1965 .

[76]  Alexander Schrijver,et al.  Relaxations of vertex packing , 1986, J. Comb. Theory B.

[77]  Stephen P. Boyd,et al.  Semidefinite Programming , 1996, SIAM Rev..

[78]  B. Mohar,et al.  Eigenvalues and the max-cut problem , 1990 .

[79]  Vojtech Rödl,et al.  The algorithmic aspects of the regularity lemma , 1992, Proceedings., 33rd Annual Symposium on Foundations of Computer Science.

[80]  Hanif D. Sherali,et al.  A Hierarchy of Relaxations Between the Continuous and Convex Hull Representations for Zero-One Programming Problems , 1990, SIAM J. Discret. Math..

[81]  David R. Karger,et al.  An Õ(n^{3/14})-Coloring Algorithm for 3-Colorable Graphs , 1997, Information Processing Letters.

[82]  Oded Schramm.  How to cage an egg , 1992 .

[83]  Robert J. Vanderbei,et al.  The Simplest Semidefinite Programs are Trivial , 1995, Math. Oper. Res..

[84]  David R. Karger,et al.  Approximate graph coloring by semidefinite programming , 1998, JACM.

[85]  M. Golummc.  Algorithmic graph theory and perfect graphs , 1980 .

[86]  David P. Williamson,et al.  .879-approximation algorithms for MAX CUT and MAX 2SAT , 1994, STOC '94.

[87]  Farid Alizadeh,et al.  Combinatorial Optimization with Semi-Definite Matrices , 1992, IPCO.

[88]  Noga Alon,et al.  Explicit Ramsey graphs and orthonormal labelings , 1994, Electron. J. Comb..

[89]  V. Chvátal.  On certain polytopes associated with graphs , 1975 .

[90]  B. S. Kashin,et al.  On systems of vectors in a Hilbert space , 1981 .

[91]  Uri Zwick,et al.  Outward rotations: a tool for rounding solutions of semidefinite programming relaxations, with applications to MAX CUT and other problems , 1999, STOC '99.

[92]  Ravi B. Boppana,et al.  Approximating maximum independent sets by excluding subgraphs , 1990, BIT.

[93]  László Lovász,et al.  Facets with fixed defect of the stable set polytope , 2000, Math. Program..

[94]  Y. D. Verdière.  On a novel graph invariant and a planarity criterion , 1990 .

[95]  Nabil Kahale,et al.  A semidefinite bound for mixing rates of Markov chains , 1996, Random Struct. Algorithms.

[96]  W. T. Tutte.  How to Draw a Graph , 1963 .

[97]  László Lovász,et al.  Normal hypergraphs and the perfect graph conjecture , 1972, Discret. Math..

[98]  László Lovász,et al.  Stable sets and polynomials , 1994, Discret. Math..

[99]  L.,et al.  On the invariance of Colin de Verdiere's graph parameter under clique sums , 2003 .

[100]  Éva Tardos,et al.  The gap between monotone and non-monotone circuit complexity is exponential , 1988, Comb..

[101]  Uriel Feige,et al.  Randomized graph products, chromatic numbers, and Lovasz j-function , 1995, STOC '95.

[102]  Satissed Now Consider.  Improved Approximation Algorithms for Maximum Cut and Satissability Problems Using Semideenite Programming , 1997 .

[103]  Lars Engebretsen,et al.  Clique Is Hard To Approximate Within , 2000 .