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43
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 1249 (13 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 ...
A spectral algorithm for envelope reduction of sparse matrices
 ACM/IEEE CONFERENCE ON SUPERCOMPUTING
, 1993
"... The problem of reordering a sparse symmetric matrix to reduce its envelope size is considered. A new spectral algorithm for computing an envelopereducing reordering is obtained by associating a Laplacian matrix with the given matrix and then sorting the components of a specified eigenvector of the ..."
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Cited by 87 (5 self)
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The problem of reordering a sparse symmetric matrix to reduce its envelope size is considered. A new spectral algorithm for computing an envelopereducing reordering is obtained by associating a Laplacian matrix with the given matrix and then sorting the components of a specified eigenvector of the Laplacian. This Laplacian eigenvector solves a continuous relaxation of a discrete problem related to envelope minimization called the minimum 2sum problem. The permutation vector computed by the spectral algorithm is a closest permutation vector to the specified Laplacian eigenvector. Numerical results show that the new reordering algorithm usually computes smaller envelope sizes than those obtained from the current standards such as the GibbsPooleStockmeyer (GPS) algorithm or the reverse CuthillMcKee (RCM) algorithm in SPARSPAK, in some cases reducing the envelope by more than a factor of two.
Indefinite Trust Region Subproblems And Nonsymmetric Eigenvalue Perturbations
, 1995
"... This paper extends the theory of trust region subproblems in two ways: (i) it allows indefinite inner products in the quadratic constraint and (ii) it uses a two sided (upper and lower bound) quadratic constraint. Characterizations of optimality are presented, which have no gap between necessity and ..."
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Cited by 73 (18 self)
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This paper extends the theory of trust region subproblems in two ways: (i) it allows indefinite inner products in the quadratic constraint and (ii) it uses a two sided (upper and lower bound) quadratic constraint. Characterizations of optimality are presented, which have no gap between necessity and sufficiency. Conditions for the existence of solutions are given in terms of the definiteness of a matrix pencil. A simple dual program is intro...
Graph Partitioning Algorithms With Applications To Scientific Computing
 Parallel Numerical Algorithms
, 1997
"... Identifying the parallelism in a problem by partitioning its data and tasks among the processors of a parallel computer is a fundamental issue in parallel computing. This problem can be modeled as a graph partitioning problem in which the vertices of a graph are divided into a specified number of su ..."
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Cited by 50 (0 self)
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Identifying the parallelism in a problem by partitioning its data and tasks among the processors of a parallel computer is a fundamental issue in parallel computing. This problem can be modeled as a graph partitioning problem in which the vertices of a graph are divided into a specified number of subsets such that few edges join two vertices in different subsets. Several new graph partitioning algorithms have been developed in the past few years, and we survey some of this activity. We describe the terminology associated with graph partitioning, the complexity of computing good separators, and graphs that have good separators. We then discuss early algorithms for graph partitioning, followed by three new algorithms based on geometric, algebraic, and multilevel ideas. The algebraic algorithm relies on an eigenvector of a Laplacian matrix associated with the graph to compute the partition. The algebraic algorithm is justified by formulating graph partitioning as a quadratic assignment p...
The algebraic theory of recombination spaces
, 2000
"... A new mathematical representation is proposed for the configuration space structure induced by recombination which we called "Pstructure". It consists of a mapping of pairs of objects to the power set of all objects in the search space. The mapping assigns to each pair of parental "g ..."
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Cited by 35 (16 self)
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A new mathematical representation is proposed for the configuration space structure induced by recombination which we called "Pstructure". It consists of a mapping of pairs of objects to the power set of all objects in the search space. The mapping assigns to each pair of parental "genotypes" the set of all recombinant genotypes obtainable from the parental ones. It is shown that this construction allows a Fourierdecomposition of fitness landscapes into a superposition of "elementary landscapes". This decomposition is analogous to the Fourier decomposition of fitness landscapes on mutation spaces. The elementary landscapes are obtained as eigenfunctions of a Laplacian operator defined for Pstructures. For binary string recombination the elementary landscapes are exactly the pspin functions (Walsh functions), i.e. the same as the elementary landscapes of the string point mutation spaces (i.e. the hypercube). This supports the notion of a strong homomorphisms between string mutation ...
Binary Partitioning, Perceptual Grouping, and Restoration with Semidefinite Programming
 IEEE Transactions on Pattern Analysis and Machine Intelligence
, 2003
"... We introduce a novel optimization method based on semidefinite programming relaxations to the field of computer vision and apply it to the combinatorial problem of minimizing quadratic functionals in binary decision variables subject to linear constraints. ..."
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Cited by 35 (6 self)
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We introduce a novel optimization method based on semidefinite programming relaxations to the field of computer vision and apply it to the combinatorial problem of minimizing quadratic functionals in binary decision variables subject to linear constraints.
Spectral Methods for Mesh Processing and Analysis
 EUROGRAPHICS 2007
, 2007
"... Spectral methods for mesh processing and analysis rely on the eigenvalues, eigenvectors, or eigenspace projections derived from appropriately defined mesh operators to carry out desired tasks. Early works in this area can be traced back to the seminal paper by Taubin in 1995, where spectral analysis ..."
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Cited by 31 (0 self)
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Spectral methods for mesh processing and analysis rely on the eigenvalues, eigenvectors, or eigenspace projections derived from appropriately defined mesh operators to carry out desired tasks. Early works in this area can be traced back to the seminal paper by Taubin in 1995, where spectral analysis of mesh geometry based on a combinatorial Laplacian aids our understanding of the lowpass filtering approach to mesh smoothing. Over the past ten years or so, the list of applications in the area of geometry processing which utilize the eigenstructures of a variety of mesh operators in different manners have been growing steadily. Many works presented so far draw parallels from developments in fields such as graph theory, computer vision, machine learning, graph drawing, numerical linear algebra, and highperformance computing. This stateoftheart report aims to provide a comprehensive survey on the spectral approach, focusing on its power and versatility in solving geometry processing problems and attempting to bridge the gap between relevant research in computer graphics and other fields. Necessary theoretical background will be provided and existing works will be classified according to different criteria — the operators or eigenstructures employed, application domains, or the dimensionality of the spectral embeddings used — and described in adequate length. Finally, despite much empirical success, there still remain many open questions pertaining to the spectral approach, which we will discuss in the report as well.
An Analysis of Spectral Envelope Reduction via Quadratic Assignment Problems
 SIAM J. Matrix Anal. Appl
, 1994
"... . A new spectral algorithm for reordering a sparse symmetric matrix to reduce its envelope size was described in [2]. The ordering is computed by associating a Laplacian matrix with the given matrix and then sorting the components of a specified eigenvector of the Laplacian. In this paper we provide ..."
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Cited by 18 (0 self)
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. A new spectral algorithm for reordering a sparse symmetric matrix to reduce its envelope size was described in [2]. The ordering is computed by associating a Laplacian matrix with the given matrix and then sorting the components of a specified eigenvector of the Laplacian. In this paper we provide an analysis of the spectral envelope reduction algorithm. We describe related 1 and 2sum problems; the former is related to the envelope size, while the latter is related to an upper bound on the work in an envelope Cholesky factorization. We formulate the latter two problems as quadratic assignment problems, and then study the 2sum problem in more detail. We obtain lower bounds on the 2sum by considering a relaxation of the problem, and then show that the spectral ordering finds a permutation matrix closest to an orthogonal matrix attaining the lower bound. This provides stronger justification of the spectral envelope reduction algorithm than previously known. The lower bound on the 2...