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43
Semidefinite Programming and Combinatorial Optimization
 Appl. Numer. Math
, 1998
"... Semidefinite Programs have recently turned out to be a powerful tool for approximating integer problems. To survey the development in this area over the last few years, the following topics are addressed in some detail. First, we investigate ways to derive semidefinite programs from discrete opti ..."
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Semidefinite Programs have recently turned out to be a powerful tool for approximating integer problems. To survey the development in this area over the last few years, the following topics are addressed in some detail. First, we investigate ways to derive semidefinite programs from discrete optimization problems. The duality theory for semidefinite programs is the key to understand algorithms to solve them. The relevant duality results are therefore summarized. The second part of the paper deals with the approximation of integer problems both in a theoretical setting, and from a computational point of view. 1 Introduction The interest in Semidefinite Programming (SDP) has been growing rapidly in the last few years. Here are some possible explanations for this sudden rise of interest. The algorithmic development of interiorpoint methods for Linear Programs indicated the potential of this approach to solve general convex problems. Semidefinite Programs are a natural generaliza...
A STRENGTHENED SDP RELAXATION via a SECOND LIFTING for the MAXCUT PROBLEM
, 1999
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Exact Bounds on the order of the maximum clique of a graph
, 2003
"... The paper reviews some of the existing exact bounds to the maximum clique of a graph and successivel presents a new upper and a newlwx, bound. The new upper bound is rank # A=2, where # A is the adjacency matrix of the compljWxyjkV graph, and derives from a formuly,G8 of the maximumclimu ..."
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Cited by 16 (0 self)
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The paper reviews some of the existing exact bounds to the maximum clique of a graph and successivel presents a new upper and a newlwx, bound. The new upper bound is rank # A=2, where # A is the adjacency matrix of the compljWxyjkV graph, and derives from a formuly,G8 of the maximumclimu problm incompl9 space. The newlwxj bound is ! 1=(1 g j # (# # )) (see text fordetail8 and improves strictl the present best lstx bound publdxWW byWil (J. Combin. Theory Ser. B 40 (1986) 113). Throughout
Semidefinite programming for discrete optimization and matrix completion problems
 Discrete Appl. Math
, 2002
"... Survey article for the proceedings of Discrete Optimization '99 where some of these results were presented as a plenary address. y ..."
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Survey article for the proceedings of Discrete Optimization '99 where some of these results were presented as a plenary address. y
Fastest mixing Markov chain on graphs with symmetries
 SIAM J. Optim
"... We show how to exploit symmetries of a graph to efficiently compute the fastest mixing Markov chain on the graph (i.e., find the transition probabilities on the edges to minimize the secondlargest eigenvalue modulus of the transition probability matrix). Exploiting symmetry can lead to significant ..."
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We show how to exploit symmetries of a graph to efficiently compute the fastest mixing Markov chain on the graph (i.e., find the transition probabilities on the edges to minimize the secondlargest eigenvalue modulus of the transition probability matrix). Exploiting symmetry can lead to significant reduction in both the number of variables and the size of matrices in the corresponding semidefinite program, thus enable numerical solution of largescale instances that are otherwise computationally infeasible. We obtain analytic or semianalytic results for particular classes of graphs, such as edgetransitive and distancetransitive graphs. We describe two general approaches for symmetry exploitation, based on orbit theory and blockdiagonalization, respectively. We also establish the connection between these two approaches. Key words. Markov chains, eigenvalue optimization, semidefinite programming, graph automorphism, group representation. 1
Spectral Mesh Processing
"... 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 work in this area can be traced back to the seminal paper by Taubin in 1995, where spectral analysis ..."
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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 work 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 fifteen years, 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 paper 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 is provided. Existing works covered are classified according to different criteria: the operators or eigenstructures employed, application domains, or the dimensionality of the spectral embeddings used. Despite much empirical success, there still remain many open questions pertaining to the spectral approach. These are discussed as we conclude the survey and provide our perspective on possible future research.
Correlation Length, Isotropy, and Metastable States
, 1997
"... A landscape is rugged if it has many local optima, if it gives rise to short adaptive walks, and if it exhibits a rapidly decreasing paircorrelation function (and hence if it has a short correlation length). The "correlation length conjecture" allows to estimate the number of metastable ..."
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A landscape is rugged if it has many local optima, if it gives rise to short adaptive walks, and if it exhibits a rapidly decreasing paircorrelation function (and hence if it has a short correlation length). The "correlation length conjecture" allows to estimate the number of metastable states from the correlation length, provided the landscape is "typical". Isotropy, originally introduced as a geometrical condition on the covariance matrix of a random field, can be reinterpreted as maximum entropy condition that lends a precise meaning to the notion of a "typical" landscape. The XYHamiltonian, which violates isotropy only to a relatively small extent, is an ideal model for investigating the influence of anisotropies. Numerical estimates for the number of local optima and predictions obtained from the correlation length conjecture indeed show deviations that increase with the extent of anisotropies in the model.
Combining Spectral Sequencing and Parallel Simulated Annealing for the MinLA Problem
 PARALLEL PROCESSING LETTERS
, 2003
"... In this paper we present and analyze new sequential and parallel heuristics to approximate the Minimum Linear Arrangement problem (MinLA). The heuristics consist in obtaining a first global solution using Spectral Sequencing and improving it locally through Simulated Annealing. In order to accele ..."
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Cited by 10 (1 self)
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In this paper we present and analyze new sequential and parallel heuristics to approximate the Minimum Linear Arrangement problem (MinLA). The heuristics consist in obtaining a first global solution using Spectral Sequencing and improving it locally through Simulated Annealing. In order to accelerate the annealing process, we present a special neighborhood distribution that tends to favor moves with high probability to be accepted. We show how to make use of this neighborhood to parallelize the Metropolis stage on distributed memory machines by mapping partitions of the input graph to processors and performing moves concurrently. The paper reports the results obtained with this new heuristic when applied to a set of large graphs, including graphs arising from finite elements methods and graphs arising from VLSI applications. Compared to other heuristics, the measurements obtained show that the new heuristic improves the solution quality, decreases the running time and o#ers an excellent speedup when ran on a commodity network made of nine personal computers.
SpectralBased MultiWay FPGA Partitioning
 IEEE Transactions on ComputerAided Design of Integrated Circuits and Systems
, 1995
"... Recent research on FPGA partitioning has focussed on finding minimum cuts between partitions without regard to the routability of the partitioned subcircuits. In this paper we develop a spectral approach to multiway partitioning in which the primary goal is to produce routable subcircuits while max ..."
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Cited by 10 (1 self)
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Recent research on FPGA partitioning has focussed on finding minimum cuts between partitions without regard to the routability of the partitioned subcircuits. In this paper we develop a spectral approach to multiway partitioning in which the primary goal is to produce routable subcircuits while maximizing FPGA device utilization. To assist the partitioner in assessing the routability of the partitioned subcircuits, we have developed a theory to predict the routability of the partitioned subcircuits prior to partitioning. Advancement over the current work is evidenced by results of experiments on the standard MCNC benchmarks. I FPGA partitioning The design flow using commercial FPGA tools involves technology mapping, placement and routing. Since FPGA devices have relatively low density, the use of multiple FPGAs is often required to implement a large circuit. A large circuit has to be decomposed or partitioned into subcircuits for a multipleFPGA realization, as shown in Fig. 1. Mod...
Complex Adaptations and the Structure of Recombination Spaces
 SCHOOL OF MATHEMATICS, UEA, NORWICH NR4 7TJ
, 1997
"... According to the Darwinian theory of evolution, adaptation results from spontaneously generated genetic variation and natural selection. Mathematical models of this process can be seen as describing a dynamics on an algebraic structure which in turn is defined by the processes which generate genetic ..."
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Cited by 10 (5 self)
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According to the Darwinian theory of evolution, adaptation results from spontaneously generated genetic variation and natural selection. Mathematical models of this process can be seen as describing a dynamics on an algebraic structure which in turn is defined by the processes which generate genetic variation (mutation and/or recombination). The theory of complex adaptive system has shown that the properties of the algebraic structure induced by mutation and recombination is more important for understanding the dynamics than the differential equations themselves. This has motivated new directions in the mathematical analysis of evolutionary models in which the algebraic properties induced by mutation and recombination are at the center of interest. In this paper we summarize some new results on the algebraic properties of recombination spaces. It is shown that the algebraic structure induced by recombination can be represented by a map from the pairs of types to the power set of the ty...