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27
Improved Approximation Algorithms for Maximum Cut and Satisfiability Problems Using Semidefinite Programming
 Journal of the ACM
, 1995
"... We present randomized approximation algorithms for the maximum cut (MAX CUT) and maximum 2satisfiability (MAX 2SAT) problems that always deliver solutions of expected value at least .87856 times the optimal value. These algorithms use a simple and elegant technique that randomly rounds the solution ..."
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Cited by 935 (14 self)
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We present randomized approximation algorithms for the maximum cut (MAX CUT) and maximum 2satisfiability (MAX 2SAT) problems that always deliver solutions of expected value at least .87856 times the optimal value. These algorithms use a simple and elegant technique that randomly rounds the solution to a nonlinear programming relaxation. This relaxation can be interpreted both as a semidefinite program and as an eigenvalue minimization problem. The best previously known approximation algorithms for these problems had performance guarantees of ...
The Dense kSubgraph Problem
 Algorithmica
, 1999
"... This paper considers the problem of computing the dense kvertex subgraph of a given graph, namely, the subgraph with the most edges. An approximation algorithm is developed for the problem, with approximation ratio O(n ffi ), for some ffi ! 1=3. 1 Introduction We study the dense ksubgraph (D ..."
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Cited by 162 (9 self)
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This paper considers the problem of computing the dense kvertex subgraph of a given graph, namely, the subgraph with the most edges. An approximation algorithm is developed for the problem, with approximation ratio O(n ffi ), for some ffi ! 1=3. 1 Introduction We study the dense ksubgraph (DkS) maximization problem, of computing the dense k vertex subgraph of a given graph. That is, on input a graph G and a parameter k, we are interested in finding a set of k vertices with maximum average degree in the subgraph induced by this set. As this problem is NPhard (say, by reduction from Clique), we consider approximation algorithms for this problem. We obtain a polynomial time algorithm that on any input (G; k) returns a subgraph of size k whose average degree is within a factor of at most n ffi from the optimum solution, where n is the number of vertices in the input graph G, and ffi ! 1=3 is some universal constant. Unfortunately, we are unable to present a complementary negati...
The Complex Structures Singular Value
, 1993
"... A tutorial introduction to the complex structured singular value (µ) is presented, with an emphasis on the mathematical aspects of µ. The µbased methods discussed here have been useful for analyzing the performance and robustness properties of linear feedback systems. Several tests ..."
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Cited by 120 (10 self)
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A tutorial introduction to the complex structured singular value (µ) is presented, with an emphasis on the mathematical aspects of µ. The µbased methods discussed here have been useful for analyzing the performance and robustness properties of linear feedback systems. Several tests
Some Applications of Laplace Eigenvalues of Graphs
 GRAPH SYMMETRY: ALGEBRAIC METHODS AND APPLICATIONS, VOLUME 497 OF NATO ASI SERIES C
, 1997
"... In the last decade important relations between Laplace eigenvalues and eigenvectors of graphs and several other graph parameters were discovered. In these notes we present some of these results and discuss their consequences. Attention is given to the partition and the isoperimetric properties of ..."
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Cited by 90 (0 self)
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In the last decade important relations between Laplace eigenvalues and eigenvectors of graphs and several other graph parameters were discovered. In these notes we present some of these results and discuss their consequences. Attention is given to the partition and the isoperimetric properties of graphs, the maxcut problem and its relation to semidefinite programming, rapid mixing of Markov chains, and to extensions of the results to infinite graphs.
Rank Minimization under LMI constraints: A Framework for Output Feedback Problems
, 1993
"... Convex optimisation techniques for solving linear matrix inequalities have been recently applied to robust statefeedback synthesis for uncertain systems. The approach does not seem to extend easily to the output feedback problem. In this paper, we propose a single framework for addressing a number ..."
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Cited by 28 (2 self)
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Convex optimisation techniques for solving linear matrix inequalities have been recently applied to robust statefeedback synthesis for uncertain systems. The approach does not seem to extend easily to the output feedback problem. In this paper, we propose a single framework for addressing a number of output feedback stabilization problems for LTI systems. This includes static output feedback stabilization, dynamic reducedorder outputfeedback stabilization, reducedorder H1 synthesis and synthesis with constant scalings. Keywords: Output feedback stabilization, linear matrix inequalities, convex optimization, H1 and synthesis. Dept. InformatiqueAutomatique, Ecole Nationale Sup'erieure de Techniques Avanc'ees, 32, Bd. Victor, 75015 Paris, France, email: elghaoui@ensta.fr. Tel: (331) 45 52 54 30, Fax: (331) 45 52 55 87. y INRIARocquencourt, BP 105, 78153 Le Chesnay Cedex, France, email: gahinet@rossini.inria.fr. 1 1 Introduction Linear Matrix Inequalities (LMI) are ineq...
A Predictor Corrector Method for Semidefinite Linear Programming
, 1995
"... In this paper we present a generalization of the predictor corrector method of linear programming problem to semidefinite linear programming problem. We consider a direction which, we show, belongs to a family of directions presented by Kojima, Shindoh and Hara, and, one of the directions analyzed b ..."
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Cited by 25 (1 self)
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In this paper we present a generalization of the predictor corrector method of linear programming problem to semidefinite linear programming problem. We consider a direction which, we show, belongs to a family of directions presented by Kojima, Shindoh and Hara, and, one of the directions analyzed by Monteiro. We show that starting with the initial complementary slackness violation of t 0 , in O(jlog( ffl t 0 )j p n) iterations of the predictor corrector method, the complementary slackness violation can be reduced to less than or equal to ffl ? 0. We also analyze a modified corrector direction in which the linear system to be solved differs from that of the predictor in only the right hand side, and obtain a similar bound. We then use this modified corrector step in an implementable method which is shown to take a total of O(jlog( ffl t 0 )j p nlog(n)) predictor and corrector steps. Key words: Linear programming, Semidefinite programming, Interior point methods, Path following, ...
Polynomial Convergence of a New Family of PrimalDual Algorithms for Semidefinite Programming
, 1996
"... This paper establishes the polynomial convergence of a new class of (feasible) primaldual interiorpoint path following algorithms for semidefinite programming (SDP) whose search directions are obtained by applying Newton method to the symmetric central path equation (P T XP ) 1=2 (P \Gamma1 ..."
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Cited by 25 (8 self)
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This paper establishes the polynomial convergence of a new class of (feasible) primaldual interiorpoint path following algorithms for semidefinite programming (SDP) whose search directions are obtained by applying Newton method to the symmetric central path equation (P T XP ) 1=2 (P \Gamma1 SP \GammaT )(P T XP ) 1=2 \Gamma I = 0; where P is a nonsingular matrix. Specifically, we show that the shortstep path following algorithm based on the Frobenius norm neighborhood and the semilongstep path following algorithm based on the operator 2norm neighborhood have O( p nL) and O(nL) iterationcomplexity bounds, respectively. When P = I, this yields the first polynomially convergent semilongstep algorithm based on a pure Newton direction. Restricting the scaling matrix P at each iteration to a certain subset of nonsingular matrices, we are able to establish an O(n 3=2 L) iterationcomplexity for the longstep path following method. The resulting subclass of search direct...
Implementation of PrimalDual Methods for Semidefinite Programming Based on Monteiro and Tsuchiya Newton Directions and their Variants
 TECHNICAL REPORT, SCHOOL INDUSTRIAL AND SYSTEMS ENGINEERING, GEORGIA TECH., ATLANTA, GA 30332
, 1997
"... Monteiro and Tsuchiya [23] have proposed two primaldual Newton directions for semidefinite programming, referred to as the MT directions, and established polynomial convergence of path following methods based on them. This paper reports some computational results on the performance of interiorpoin ..."
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Cited by 20 (3 self)
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Monteiro and Tsuchiya [23] have proposed two primaldual Newton directions for semidefinite programming, referred to as the MT directions, and established polynomial convergence of path following methods based on them. This paper reports some computational results on the performance of interiorpoint predictorcorrector methods based on the MT directions and a variant of these directions, called the SChMT direction. We discuss how to compute these directions efficiently and derive their corresponding computational complexities. A main feature of our analysis is that computational formulae for these directions are derived from a unified point of view which entirely avoids the use of Kronecker product. Using this unified approach, we also present schemes to compute the AlizadehHaeberlyOverton (AHO) direction, the NesterovTodd direction and the HRVW/KSH/M direction with computational complexities (for dense problems) better than previously reported in the literature. Our computational...
A Unified Analysis for a Class of LongStep PrimalDual PathFollowing InteriorPoint Algorithms for Semidefinite Programming
 MATH. PROGRAMMING
, 1998
"... We present a unified analysis for a class of longstep primaldual pathfollowing algorithms for semidefinite programming whose search directions are obtained through linearization of the symmetrized equation of the central path HP (XS) j [P XSP \Gamma1 + (PXSP \Gamma1 ) T ]=2 = ¯I, introduce ..."
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Cited by 18 (0 self)
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We present a unified analysis for a class of longstep primaldual pathfollowing algorithms for semidefinite programming whose search directions are obtained through linearization of the symmetrized equation of the central path HP (XS) j [P XSP \Gamma1 + (PXSP \Gamma1 ) T ]=2 = ¯I, introduced by Zhang. At an iterate (X; S), we choose a scaling matrix P from the class of nonsingular matrices P such that PXSP \Gamma1 is symmetric. This class of matrices includes the three wellknown choices, namely: P = S 1=2 and P = X \Gamma1=2 proposed by Monteiro, and the matrix P corresponding to the NesterovTodd direction. We show that within the class of algorithms studied in this paper, the one based on the NesterovTodd direction has the lowest possible iterationcomplexity bound that can provably be derived from our analysis. More specifically, its iterationcomplexity bound is of the same order as that of the corresponding longstep primaldual pathfollowing algorithm for linear...