Approximation Algorithms and Semidefinite Programming

Semidefinite programs constitute one of the largest classes of optimization problems that can be solved with reasonable efficiency - both in theory and practice. They play a key role in a variety of research areas, such as combinatorial optimization, approximation algorithms, computational complexity, graph theory, geometry, real algebraic geometry and quantum computing. This book is an introduction to selected aspects of semidefinite programming and its use in approximation algorithms. It covers the basics but also a significant amount of recent and more advanced material. There are many computational problems, such as MAXCUT, for which one cannot reasonably expect to obtain an exact solution efficiently, and in such case, one has to settle for approximate solutions. For MAXCUT and its relatives, exciting recent results suggest that semidefinite programming is probably the ultimate tool. Indeed, assuming the Unique Games Conjecture, a plausible but as yet unproven hypothesis, it was shown that for these problems, known algorithms based on semidefinite programming deliver the best possible approximation ratios among all polynomial-time algorithms. This book follows the semidefinite side of these developments, presenting some of the main ideas behind approximation algorithms based on semidefinite programming. It develops the basic theory of semidefinite programming, presents one of the known efficient algorithms in detail, and describes the principles of some others. It also includes applications, focusing on approximation algorithms.

[1]  Philip Wolfe,et al.  An algorithm for quadratic programming , 1956 .

[2]  Claude E. Shannon,et al.  The zero error capacity of a noisy channel , 1956, IRE Trans. Inf. Theory.

[3]  T. Motzkin,et al.  Maxima for Graphs and a New Proof of a Theorem of Turán , 1965, Canadian Journal of Mathematics.

[4]  R. Rietz.  A proof of the Grothendieck inequality , 1974 .

[5]  David S. Johnson,et al.  Some Simplified NP-Complete Graph Problems , 1976, Theor. Comput. Sci..

[6]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[7]  G. Stewart,et al.  Estimating the Largest Eigenvalue of a Positive Definite Matrix , 1979 .

[8]  L. Lovász,et al.  Geometric Algorithms and Combinatorial Optimization , 1981 .

[9]  Gene H. Golub,et al.  Matrix computations , 1983 .

[10]  J. Spencer.  Six standard deviations suffice , 1985 .

[11]  Katta G. Murty,et al.  Some NP-complete problems in quadratic and nonlinear programming , 1987, Math. Program..

[12]  J. Spencer.  Ten lectures on the probabilistic method , 1987 .

[13]  A. Peressini,et al.  The Mathematics Of Nonlinear Programming , 1988 .

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

[15]  N. Higham.  Analysis of the Cholesky Decomposition of a Semi-definite Matrix , 1990 .

[16]  Noga Alon,et al.  Multilinear polynomials and Frankl-Ray-Chaudhuri-Wilson type intersection theorems , 1991, J. Comb. Theory, Ser. A.

[17]  Henryk Wozniakowski,et al.  Estimating the Largest Eigenvalue by the Power and Lanczos Algorithms with a Random Start , 1992, SIAM J. Matrix Anal. Appl..

[18]  Avrim Blum,et al.  New approximation algorithms for graph coloring , 1994, JACM.

[19]  Vojtech Rödl,et al.  The Algorithmic Aspects of the Regularity Lemma , 1994, J. Algorithms.

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

[21]  Marek Karpinski,et al.  Polynomial time approximation schemes for dense instances of NP-hard problems , 1995, STOC '95.

[22]  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.

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

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

[25]  Hsueh-I Lu,et al.  Efficient approximation algorithms for semidefinite programs arising from MAX CUT and COLORING , 1996, STOC '96.

[26]  Robert J. Vanderbei,et al.  An Interior-Point Method for Semidefinite Programming , 1996, SIAM J. Optim..

[27]  A. Grothendieck.  Résumé de la théorie métrique des produits tensoriels topologiques , 1996 .

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

[29]  Renato D. C. Monteiro,et al.  Primal-Dual Path-Following Algorithms for Semidefinite Programming , 1997, SIAM J. Optim..

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

[31]  Shinji Hara,et al.  Interior-Point Methods for the Monotone Semidefinite Linear Complementarity Problem in Symmetric Matrices , 1997, SIAM J. Optim..

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

[33]  Y. Nesterov.  Semidefinite relaxation and nonconvex quadratic optimization , 1998 .

[34]  Uriel Feige,et al.  Zero Knowledge and the Chromatic Number , 1998, J. Comput. Syst. Sci..

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

[36]  Tomás Feder,et al.  The Computational Structure of Monotone Monadic SNP and Constraint Satisfaction: A Study through Datalog and Group Theory , 1999, SIAM J. Comput..

[37]  Tamás Terlaky,et al.  On maximization of quadratic form over intersection of ellipsoids with common center , 1999, Math. Program..

[38]  Alan M. Frieze,et al.  Quick Approximation to Matrices and Applications , 1999, Comb..

[39]  Sanjeev Mahajan,et al.  Derandomizing Approximation Algorithms Based on Semidefinite Programming , 1999, SIAM J. Comput..

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

[41]  Michael J. Todd,et al.  Path-Following Methods , 2000 .

[42]  Noga Alon,et al.  Bipartite Subgraphs and the Smallest Eigenvalue , 2000, Combinatorics, Probability and Computing.

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

[44]  Marek Karpinski,et al.  Improved approximation of Max-Cut on graphs of bounded degree , 2002, J. Algorithms.

[45]  Etienne de Klerk,et al.  On Copositive Programming and Standard Quadratic Optimization Problems , 2000, J. Glob. Optim..

[46]  Nathan Linial,et al.  On the Hardness of Approximating the Chromatic Number , 2000, Comb..

[47]  Arkadi Nemirovski,et al.  Lectures on modern convex optimization - analysis, algorithms, and engineering applications , 2001, MPS-SIAM series on optimization.

[48]  Johan Håstad,et al.  Some optimal inapproximability results , 2001, JACM.

[49]  Uriel Feige,et al.  On the optimality of the random hyperplane rounding technique for MAX CUT , 2002, Random Struct. Algorithms.

[50]  Uri Zwick,et al.  Improved Rounding Techniques for the MAX 2-SAT and MAX DI-CUT Problems , 2002, IPCO.

[51]  Subhash Khot,et al.  On the power of unique 2-prover 1-round games , 2002, Proceedings 17th IEEE Annual Conference on Computational Complexity.

[52]  Franz Rendl,et al.  Semidefinite programming and integer programming , 2002 .

[53]  Andrei A. Bulatov,et al.  A dichotomy theorem for constraints on a three-element set , 2002, The 43rd Annual IEEE Symposium on Foundations of Computer Science, 2002. Proceedings..

[54]  László Lovász,et al.  Semidefinite Programs and Combinatorial Optimization , 2003 .

[55]  Monique Laurent,et al.  A Comparison of the Sherali-Adams, Lovász-Schrijver, and Lasserre Relaxations for 0-1 Programming , 2003, Math. Oper. Res..

[56]  Guy Kindler,et al.  Optimal inapproximability results for MAX-CUT and other 2-variable CSPs? , 2004, 45th Annual IEEE Symposium on Foundations of Computer Science.

[57]  József Beck,et al.  Geometric Discrepancy Theory Anduniform Distribution , 2004, Handbook of Discrete and Computational Geometry, 2nd Ed..

[58]  Moses Charikar,et al.  Maximizing quadratic programs: extending Grothendieck's inequality , 2004, 45th Annual IEEE Symposium on Foundations of Computer Science.

[59]  V. Milman,et al.  Functional analysis : an introduction , 2004 .

[60]  Noga Alon,et al.  Approximating the cut-norm via Grothendieck's inequality , 2004, STOC '04.

[61]  Sanjeev Arora,et al.  New approximation guarantee for chromatic number , 2006, STOC '06.

[62]  Stephen P. Boyd,et al.  Convex Optimization , 2004, Algorithms and Theory of Computation Handbook.

[63]  Michael Langberg,et al.  The RPR2 rounding technique for semidefinite programs , 2006, J. Algorithms.

[64]  Elchanan Mossel,et al.  Conditional hardness for approximate coloring , 2005, STOC '06.

[65]  Bernd Gärtner,et al.  Understanding and Using Linear Programming (Universitext) , 2006 .

[66]  Subhash Khot,et al.  Better Inapproximability Results for MaxClique, Chromatic Number and Min-3Lin-Deletion , 2006, ICALP.

[67]  David Zuckerman,et al.  Electronic Colloquium on Computational Complexity, Report No. 100 (2005) Linear Degree Extractors and the Inapproximability of MAX CLIQUE and CHROMATIC NUMBER , 2005 .

[68]  Eden Chlamtác,et al.  Approximation Algorithms Using Hierarchies of Semidefinite Programming Relaxations , 2007, 48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07).

[69]  Satyen Kale.  Efficient algorithms using the multiplicative weights update method , 2007 .

[70]  Per Austrin.  Towards Sharp Inapproximability For Any 2-CSP , 2007, 48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07).

[71]  Elad Hazan,et al.  Sparse Approximate Solutions to Semidefinite Programs , 2008, LATIN.

[72]  Prasad Raghavendra,et al.  Optimal algorithms and inapproximability results for every CSP? , 2008, STOC.

[73]  Ryan O'Donnell,et al.  An optimal sdp algorithm for max-cut, and equally optimal long code tests , 2008, STOC.

[74]  Kenneth L. Clarkson,et al.  Coresets, sparse greedy approximation, and the Frank-Wolfe algorithm , 2008, SODA '08.

[75]  Frank Vallentin,et al.  Lecture notes: Semidefinite programs and harmonic analysis , 2008, 0809.2017.

[76]  Elchanan Mossel,et al.  Approximation Resistant Predicates from Pairwise Independence , 2008, 2008 23rd Annual IEEE Conference on Computational Complexity.

[77]  Konstantin Makarychev,et al.  Quadratic forms on graphs and their applications , 2008 .

[78]  P. Raghavendra,et al.  Approximating np-hard problems efficient algorithms and their limits , 2009 .

[79]  Subhash Khot,et al.  SDP gaps and UGC-hardness for MAXCUTGAIN , 2006, IEEE Annual Symposium on Foundations of Computer Science.

[80]  Madhur Tulsiani.  CSP gaps and reductions in the lasserre hierarchy , 2009, STOC '09.

[81]  J. Lasserre.  Moments, Positive Polynomials And Their Applications , 2009 .

[82]  Prasad Raghavendra,et al.  Towards computing the Grothendieck constant , 2009, SODA.

[83]  Prasad Raghavendra,et al.  Integrality Gaps for Strong SDP Relaxations of UNIQUE GAMES , 2009, 2009 50th Annual IEEE Symposium on Foundations of Computer Science.

[84]  Aleksandar Nikolov,et al.  Limits of Approximation Algorithms: PCPs and Unique Games (DIMACS Tutorial Lecture Notes) , 2010, ArXiv.

[85]  Vojtech Rödl,et al.  Quasi-Randomness and Algorithmic Regularity for Graphs with General Degree Distributions , 2007, SIAM J. Comput..

[86]  Franz Rendl,et al.  Copositive programming motivated bounds on the stability and the chromatic numbers , 2009, Math. Program..

[87]  Sanjeev Arora,et al.  Subexponential Algorithms for Unique Games and Related Problems , 2010, 2010 IEEE 51st Annual Symposium on Foundations of Computer Science.

[88]  David Steurer,et al.  Fast SDP algorithms for constraint satisfaction problems , 2010, SODA '10.

[89]  David P. Williamson,et al.  The Design of Approximation Algorithms , 2011 .

[90]  Subhash Khot.  Inapproximability of NP-complete Problems, Discrete Fourier Analysis, and Geometry , 2011 .

[91]  Aleksandar Nikolov,et al.  Tight hardness results for minimizing discrepancy , 2011, SODA '11.

[92]  Madhur Tulsiani,et al.  Convex Relaxations and Integrality Gaps , 2012 .

[93]  J. Lasserre,et al.  Handbook on Semidefinite, Conic and Polynomial Optimization , 2012 .

[94]  Sanjeev Arora,et al.  The Multiplicative Weights Update Method: a Meta-Algorithm and Applications , 2012, Theory Comput..