Results 1  10
of
99
Quick Approximation to Matrices and Applications
, 1998
"... We give algorithms to find the following simply described approximation to a given matrix. Given an m \Theta n matrix A with entries between say1 and 1, and an error parameter ffl between 0 and 1, we find a matrix D (implicitly) which is the sum of O(1=ffl 2 ) simple rank 1 matrices so that the ..."
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Cited by 122 (5 self)
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We give algorithms to find the following simply described approximation to a given matrix. Given an m \Theta n matrix A with entries between say1 and 1, and an error parameter ffl between 0 and 1, we find a matrix D (implicitly) which is the sum of O(1=ffl 2 ) simple rank 1 matrices so that the sum of entries of any submatrix (among the 2 m+n ) of (A \Gamma D) is at most fflmn in absolute value. Our algorithm takes time dependent only on ffl and the allowed probability of failure (not on m;n). We draw on two lines of research to develop the algorithms: one is built around the fundamental Regularity Lemma of Szemerédi in Graph Theory and the constructive version of Alon, Duke, Leffman, Rödl and Yuster. The second one is from the papers of Arora, Karger and Karpinski, Fernandez de la Vega and most directly Goldwasser, Goldreich and Ron who develop approximation algorithms for a set of graph problems, typical of which is the maximum cut problem. ?From our matrix approximation, the...
Exact algorithms for NPhard problems: A survey
 Combinatorial Optimization  Eureka, You Shrink!, LNCS
"... Abstract. We discuss fast exponential time solutions for NPcomplete problems. We survey known results and approaches, we provide pointers to the literature, and we discuss several open problems in this area. The list of discussed NPcomplete problems includes the travelling salesman problem, schedu ..."
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Cited by 118 (3 self)
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Abstract. We discuss fast exponential time solutions for NPcomplete problems. We survey known results and approaches, we provide pointers to the literature, and we discuss several open problems in this area. The list of discussed NPcomplete problems includes the travelling salesman problem, scheduling under precedence constraints, satisfiability, knapsack, graph coloring, independent sets in graphs, bandwidth of a graph, and many more. 1
A New Rounding Procedure for the Assignment Problem with Applications to Dense Graph Arrangement Problems
, 2001
"... We present a randomized procedure for rounding fractional perfect matchings to (integral) matchings. If the original fractional matching satis es any linear inequality, then with high probability, the new matching satis es that linear inequality in an approximate sense. This extends the wellkn ..."
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Cited by 78 (3 self)
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We present a randomized procedure for rounding fractional perfect matchings to (integral) matchings. If the original fractional matching satis es any linear inequality, then with high probability, the new matching satis es that linear inequality in an approximate sense. This extends the wellknown LP rounding procedure of Raghavan and Thompson, which is usually used to round fractional solutions of linear programs.
Solving Large Quadratic Assignment Problems on Computational Grids
, 2000
"... The quadratic assignment problem (QAP) is among the hardest combinatorial optimization problems. Some instances of size n = 30 have remained unsolved for decades. The solution of these problems requires both improvements in mathematical programming algorithms and the utilization of powerful computat ..."
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Cited by 72 (6 self)
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The quadratic assignment problem (QAP) is among the hardest combinatorial optimization problems. Some instances of size n = 30 have remained unsolved for decades. The solution of these problems requires both improvements in mathematical programming algorithms and the utilization of powerful computational platforms. In this article we describe a novel approach to solve QAPs using a stateoftheart branchandbound algorithm running on a federation of geographically distributed resources known as a computational grid. Solution of QAPs of unprecedented complexity, including the nug30, kra30b, and tho30 instances, is reported.
Semidefinite Programming Relaxations For The Quadratic Assignment Problem
, 1998
"... Semidefinite programming (SDP) relaxations for the quadratic assignment problem (QAP) are derived using the dual of the (homogenized) Lagrangian dual of appropriate equivalent representations of QAP. These relaxations result in the interesting, special, case where only the dual problem of the SDP re ..."
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Cited by 72 (25 self)
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Semidefinite programming (SDP) relaxations for the quadratic assignment problem (QAP) are derived using the dual of the (homogenized) Lagrangian dual of appropriate equivalent representations of QAP. These relaxations result in the interesting, special, case where only the dual problem of the SDP relaxation has strict interior, i.e. the Slater constraint qualification always fails for the primal problem. Although there is no duality gap in theory, this indicates that the relaxation cannot be solved in a numerically stable way. By exploring the geometrical structure of the relaxation, we are able to find projected SDP relaxations. These new relaxations, and their duals, satisfy the Slater constraint qualification, and so can be solved numerically using primaldual interiorpoint methods. For one of our models, a preconditioned conjugate gradient method is used for solving the large linear systems which arise when finding the Newton direction. The preconditioner is found by exploiting th...
Edward,“Guided Local Search
, 1995
"... Abstract Combinatorial explosion problem is a well known phenomenon that prevents complete algorithms from solving many reallife combinatorial optimization problems. In many situations, heuristic search methods are needed. This chapter describes the principles of Guided Local Search (GLS) and Fast ..."
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Cited by 57 (6 self)
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Abstract Combinatorial explosion problem is a well known phenomenon that prevents complete algorithms from solving many reallife combinatorial optimization problems. In many situations, heuristic search methods are needed. This chapter describes the principles of Guided Local Search (GLS) and Fast Local Search (FLS) and surveys their applications. GLS is a penaltybased metaheuristic algorithm that sits on top of other local search algorithms, with the aim to improve their efficiency and robustness. FLS is a way of reducing the size of the neighbourhood to improve the efficiency of local search. The chapter also provides guidance for implementing and using GLS and FLS. Four problems, representative of general application categories, are examined with detailed information provided on how to build a GLSbased method in each case.
On Lagrangian relaxation of quadratic matrix constraints
 SIAM J. Matrix Anal. Appl
, 2000
"... Abstract. Quadratically constrained quadratic programs (QQPs) play an important modeling role for many diverse problems. These problems are in general NP hard and numerically intractable. Lagrangian relaxations often provide good approximate solutions to these hard problems. Such relaxations are equ ..."
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Cited by 46 (17 self)
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Abstract. Quadratically constrained quadratic programs (QQPs) play an important modeling role for many diverse problems. These problems are in general NP hard and numerically intractable. Lagrangian relaxations often provide good approximate solutions to these hard problems. Such relaxations are equivalent to semidefinite programming relaxations. For several special cases of QQP, e.g., convex programs and trust region subproblems, the Lagrangian relaxation provides the exact optimal value, i.e., there is a zero duality gap. However, this is not true for the general QQP, or even the QQP with two convex constraints, but a nonconvex objective. In this paper we consider a certain QQP where the quadratic constraints correspond to the matrix orthogonality condition XXT = I. For this problem we show that the Lagrangian dual based on relaxing the constraints XXT = I and the seemingly redundant constraints XT X = I has a zero duality gap. This result has natural applications to quadratic assignment and graph partitioning problems, as well as the problem of minimizing the weighted sum of the largest eigenvalues of a matrix. We also show that the technique of relaxing quadratic matrix constraints can be used to obtain a strengthened semidefinite relaxation for the maxcut problem. Key words. Lagrangian relaxations, quadratically constrained quadratic programs, semidefinite programming, quadratic assignment, graph partitioning, maxcut problems
A recipe for semidefinite relaxation for 01 quadratic programming
 JOURNAL OF GLOBAL OPTIMIZATION
, 1995
"... We review various relaxations of (0,1)quadratic programming problems. These include semidefinite programs, parametric trust region problems and concave quadratic maximization. All relaxations that we consider lead to efficiently solvable problems. The main contributions of the paper are the followi ..."
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Cited by 44 (7 self)
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We review various relaxations of (0,1)quadratic programming problems. These include semidefinite programs, parametric trust region problems and concave quadratic maximization. All relaxations that we consider lead to efficiently solvable problems. The main contributions of the paper are the following. Using Lagrangian duality, we prove equivalence of the relaxations in a unified and simple way. Some of these equivalences have been known previously, but our approach leads to short and transparent proofs. Moreover we extend the approach to the case of equality constrained problems by taking the squared linear constraints into the objective function. We show how this technique can be applied to the Quadratic Assignment Problem, the Graph Partition Problem and the MaxClique Problem. Finally we show our relaxation to be best possible among all quadratic majorants with zero trace.
A New Bound for the Quadratic Assignment Problem Based on Convex Quadratic Programming
 MATHEMATICAL PROGRAMMING
, 1999
"... We describe a new convex quadratic programming bound for the quadratic assignment problem (QAP). The construction of the bound uses a semidefinite programming representation of a basic eigenvalue bound for QAP. The new bound dominates the wellknown projected eigenvalue bound, and appears to be comp ..."
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Cited by 34 (3 self)
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We describe a new convex quadratic programming bound for the quadratic assignment problem (QAP). The construction of the bound uses a semidefinite programming representation of a basic eigenvalue bound for QAP. The new bound dominates the wellknown projected eigenvalue bound, and appears to be competitive with existing bounds in the tradeoff between bound quality and computational effort.