Download or read book Fast Approximation Algorithms for Graph Partitioning Using Spectral and Semidefinite Programming Techniques written by Lorenzo Orecchia and published by . This book was released on 2011 with total page 210 pages. Available in PDF, EPUB and Kindle. Book excerpt: Graph partitioning problems are a central topic of research in the study of approximation algorithms. They are of interest to theoretical computer scientists for their far-reaching connections to spectral graph theory, metric embeddings and inapproximability theory. And they are also important for many practitioners, as algorithms for graph partitioning are often fundamental primitives in the solution of other problems, such as image segmentation, clustering and social-network analysis. While many theoretical approximation algorithms exist for graph partitioning, they often rely on multicommodity-flow computations that run quadratic time in the worst case and are too time-consuming for the massive graphs that are prevalent in today's applications. In this thesis, we study the design of approximation algorithms that yield strong approximation ratios, while running in subquadratic time and relying on computational procedures that are often fast in practice. Our algorithms employ spectral and s-t flow operations to explore the cuts of a graph in a very efficient way. A crucial ingredient in their design is the usage of random walks that capture the sparse cuts of a graph better than single eigenvectors. The analysis of our methods is particularly simple, as it relies on a semidefinite programming formulation of the graph partitioning problem of choice. Indeed, we can develop our algorithms as primal-dual methods for solving a semidefinite program and show that certain random walks arise naturally from this approach.

Download or read book Spectral Algorithms written by Ravindran Kannan and published by Now Publishers Inc. This book was released on 2009 with total page 153 pages. Available in PDF, EPUB and Kindle. Book excerpt: Spectral methods refer to the use of eigenvalues, eigenvectors, singular values and singular vectors. They are widely used in Engineering, Applied Mathematics and Statistics. More recently, spectral methods have found numerous applications in Computer Science to "discrete" as well as "continuous" problems. Spectral Algorithms describes modern applications of spectral methods, and novel algorithms for estimating spectral parameters. The first part of the book presents applications of spectral methods to problems from a variety of topics including combinatorial optimization, learning and clustering. The second part of the book is motivated by efficiency considerations. A feature of many modern applications is the massive amount of input data. While sophisticated algorithms for matrix computations have been developed over a century, a more recent development is algorithms based on "sampling on the fly" from massive matrices. Good estimates of singular values and low rank approximations of the whole matrix can be provably derived from a sample. The main emphasis in the second part of the book is to present these sampling methods with rigorous error bounds. It also presents recent extensions of spectral methods from matrices to tensors and their applications to some combinatorial optimization problems.

Download or read book From Graphs to Matrices and Back written by Aleksander Mądry and published by . This book was released on 2011 with total page 192 pages. Available in PDF, EPUB and Kindle. Book excerpt: The growing need to deal efficiently with massive computing tasks prompts us to consider the following question: How well can we solve fundamental optimization problems if our algorithms have to run really quickly? The motivation for the research presented in this thesis stems from addressing the above question in the context of algorithmic graph theory. To pursue this direction, we develop a toolkit that combines a diverse set of modern algorithmic techniques, including sparsification, low-stretch spanning trees, the multiplicative-weights-update method, dynamic graph algorithms, fast Laplacian system solvers, and tools of spectral graph theory. Using this toolkit, we obtain improved algorithms for several basic graph problems including: -- The Maximum s-t Flow and Minimum s-t Cut Problems. We develop a new approach to computing (1 - [epsilon])-approximately maximum s-t flow and (1 + [epsilon])-approximately minimum s-t cut in undirected graphs that gives the fastest known algorithms for these tasks. These algorithms are the first ones to improve the long-standing bound of O(n3/2') running time on sparse graphs; -- Multicommodity Flow Problems. We set forth a new method of speeding up the existing approximation algorithms for multicommodity flow problems, and use it to obtain the fastest-known (1 - [epsilon])-approximation algorithms for these problems. These results improve upon the best previously known bounds by a factor of roughly [omega](m/n), and make the resulting running times essentially match the [omega](mn) "flow-decomposition barrier" that is a natural obstacle to all the existing approaches; -- " Undirected (Multi-)Cut-Based Minimization Problems. We develop a general framework for designing fast approximation algorithms for (multi-)cutbased minimization problems in undirected graphs. Applying this framework leads to the first algorithms for several fundamental graph partitioning primitives, such as the (generalized) sparsest cut problem and the balanced separator problem, that run in close to linear time while still providing polylogarithmic approximation guarantees; -- The Asymmetric Traveling Salesman Problem. We design an O( )- approximation algorithm for the classical problem of combinatorial optimization: the asymmetric traveling salesman problem. This is the first asymptotic improvement over the long-standing approximation barrier of e(log n) for this problem; -- Random Spanning Tree Generation. We improve the bound on the time needed to generate an uniform random spanning tree of an undirected graph.

Download or read book Fast Approximate Graph Partitioning Algorithms written by International Business Machines Corporation. Research Division and published by . This book was released on 1998 with total page 28 pages. Available in PDF, EPUB and Kindle. Book excerpt: Abstract: "We study graph partitioning problems on graphs with edge capacities and vertex weights. The problems of b-balanced cuts and k-balanced partitions are unified into a new problem called minimum capacity p-separators. A p-separator is a subset of edges whose removal partitions the vertex set into connected components such that the sum of the vertex weights in each component is at most p times the weight of the graph. We present a new and simple O(log n)-approximation algorithm for minimum capacity p-separators which is based on spreading metrics yielding an O(log n)-approximation algorithm both for b-balanced cuts and k-balanced partitions. In particular, this result improves the previous best known approximation factor for k-balanced partitions in undirected graphs by a factor of O(log k). We enhance these results by presenting a version of the algorithm that obtains an O(log OPT)-approximation factor. The algorithm is based on a technique called spreading metrics that enables us to formulate directly the minimum capacity p-separator problem as an integer program. We also introduce a generalization called the simultaneous separator problem, where the goal is to find a minimum capacity subset of edges that separates a given collection of subsets simultaneously. We extend our results to directed graphs for values of p [> or =] 1/2. We conclude with an efficient algorithm for computing an optimal spreading metric for p-separators. This yields more efficient algorithms for computing b-balanced cuts than were previously known."

Download or read book Graph Mining written by Deepayan Chakrabarti and published by Morgan & Claypool Publishers. This book was released on 2012-10-01 with total page 209 pages. Available in PDF, EPUB and Kindle. Book excerpt: What does the Web look like? How can we find patterns, communities, outliers, in a social network? Which are the most central nodes in a network? These are the questions that motivate this work. Networks and graphs appear in many diverse settings, for example in social networks, computer-communication networks (intrusion detection, traffic management), protein-protein interaction networks in biology, document-text bipartite graphs in text retrieval, person-account graphs in financial fraud detection, and others. In this work, first we list several surprising patterns that real graphs tend to follow. Then we give a detailed list of generators that try to mirror these patterns. Generators are important, because they can help with "what if" scenarios, extrapolations, and anonymization. Then we provide a list of powerful tools for graph analysis, and specifically spectral methods (Singular Value Decomposition (SVD)), tensors, and case studies like the famous "pageRank" algorithm and the "HITS" algorithm for ranking web search results. Finally, we conclude with a survey of tools and observations from related fields like sociology, which provide complementary viewpoints. Table of Contents: Introduction / Patterns in Static Graphs / Patterns in Evolving Graphs / Patterns in Weighted Graphs / Discussion: The Structure of Specific Graphs / Discussion: Power Laws and Deviations / Summary of Patterns / Graph Generators / Preferential Attachment and Variants / Incorporating Geographical Information / The RMat / Graph Generation by Kronecker Multiplication / Summary and Practitioner's Guide / SVD, Random Walks, and Tensors / Tensors / Community Detection / Influence/Virus Propagation and Immunization / Case Studies / Social Networks / Other Related Work / Conclusions

Download or read book Graph Partitioning written by Charles-Edmond Bichot and published by John Wiley & Sons. This book was released on 2013-01-24 with total page 301 pages. Available in PDF, EPUB and Kindle. Book excerpt: Graph partitioning is a theoretical subject with applications in many areas, principally: numerical analysis, programs mapping onto parallel architectures, image segmentation, VLSI design. During the last 40 years, the literature has strongly increased and big improvements have been made. This book brings together the knowledge accumulated during many years to extract both theoretical foundations of graph partitioning and its main applications.

Download or read book Algorithm Engineering written by Lasse Kliemann and published by Springer. This book was released on 2016-11-10 with total page 428 pages. Available in PDF, EPUB and Kindle. Book excerpt: Algorithm Engineering is a methodology for algorithmic research that combines theory with implementation and experimentation in order to obtain better algorithms with high practical impact. Traditionally, the study of algorithms was dominated by mathematical (worst-case) analysis. In Algorithm Engineering, algorithms are also implemented and experiments conducted in a systematic way, sometimes resembling the experimentation processes known from fields such as biology, chemistry, or physics. This helps in counteracting an otherwise growing gap between theory and practice.

Download or read book Approximation Algorithms for New Graph Partitioning and Facility Location Problems written by Zoya Svitkina and published by . This book was released on 2007 with total page 260 pages. Available in PDF, EPUB and Kindle. Book excerpt: In applications as diverse as data placement in peer-to-peer systems, control of epidemic outbreaks, and routing in sensor networks, the fundamental questions can be abstracted as problems in combinatorial optimization. However, many of these problems are NP-hard, which makes it unlikely that exact polynomial-time algorithms for them exist. Approximation algorithms are designed to circumvent this difficulty, by finding provably near-optimal solutions in polynomial time. This thesis introduces a number of new combinatorial optimization problems that arise from various applications and proposes approximation algorithms for them. These problems fall into two general areas: graph partitioning and facility location. The first problem that we introduce is the unbalanced graph cut problem. Here the goal is to find a graph cut, minimizing the size of one of the sides, while also respecting an upper bound on the number of edges cut. We develop two bicriteria approximation algorithms for this problem using the technique of Lagrangian relaxation, and a different algorithm for its maximization version. The other graph partitioning problem that we introduce and study is the min-max multiway cut problem. It aims to partition a graph into multiple components, minimizing the maximum number of edges coming out of any component. We present an approximation algorithm for this problem which uses unbalanced cuts as well as the greedy technique. In the second part of the thesis, we study two generalizations of the facility location problem, which aims to open facilities, assigning clients to them, in order to minimize the facility opening costs and the connection costs. In the facility location with hierarchical facility costs problem, the facility costs are more general, and depend on the set of assigned clients. Our algorithm, based on the local search technique, uses two new local improvement operations, achieving a constant-factor approximation guarantee. The second generalization is the load-balanced facility location problem, which specifies a lower bound for the number of clients assigned to an open facility. We give the first true constant-factor approximation algorithm, which uses a reduction to the capacitated facility location problem. The thesis is concluded with related open problems and directions for future research. (Abstract).

Download or read book Approximation Randomization and Combinatorial Optimization Algorithms and Techniques written by Josep Diaz and published by Springer Science & Business Media. This book was released on 2006-08-11 with total page 532 pages. Available in PDF, EPUB and Kindle. Book excerpt: This is the joint refereed proceedings of the 9th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2006 and the 10th International Workshop on Randomization and Computation, RANDOM 2006. The book presents 44 carefully reviewed and revised full papers. Among the topics covered are design and analysis of approximation algorithms, hardness of approximation problems, small spaces and data streaming algorithms, embeddings and metric space methods, and more.

Download or read book Faster Algorithms Via Approximation Theory written by Sushant Sachdeva and published by . This book was released on 2014-03-28 with total page 108 pages. Available in PDF, EPUB and Kindle. Book excerpt: Faster Algorithms via Approximation Theory illustrates how classical and modern techniques from approximation theory play a crucial role in obtaining results that are relevant to the emerging theory of fast algorithms. The key lies in the fact that such results imply faster ways to approximate primitives such as products of matrix functions with vectors and, to compute matrix eigenvalues and eigenvectors, which are fundamental to many spectral algorithms. The first half of the book is devoted to the ideas and results from approximation theory that are central, elegant, and may have wider applicability in theoretical computer science. These include not only techniques relating to polynomial approximations but also those relating to approximations by rational functions and beyond. The remaining half illustrates a variety of ways that these results can be used to design fast algorithms. Faster Algorithms via Approximation Theory is self-contained and should be of interest to researchers and students in theoretical computer science, numerical linear algebra, and related areas.

Download or read book Graph Partitioning and Graph Clustering written by David A. Bader and published by American Mathematical Soc.. This book was released on 2013-03-18 with total page 258 pages. Available in PDF, EPUB and Kindle. Book excerpt: Graph partitioning and graph clustering are ubiquitous subtasks in many applications where graphs play an important role. Generally speaking, both techniques aim at the identification of vertex subsets with many internal and few external edges. To name only a few, problems addressed by graph partitioning and graph clustering algorithms are: What are the communities within an (online) social network? How do I speed up a numerical simulation by mapping it efficiently onto a parallel computer? How must components be organized on a computer chip such that they can communicate efficiently with each other? What are the segments of a digital image? Which functions are certain genes (most likely) responsible for? The 10th DIMACS Implementation Challenge Workshop was devoted to determining realistic performance of algorithms where worst case analysis is overly pessimistic and probabilistic models are too unrealistic. Articles in the volume describe and analyze various experimental data with the goal of getting insight into realistic algorithm performance in situations where analysis fails.

Download or read book Approximation Randomization and Combinatorial Optimization Algorithms and Techniques written by Prasad Raghavendra and published by Springer. This book was released on 2013-08-16 with total page 728 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book constitutes the proceedings of the 16th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2013, and the 17th International Workshop on Randomization and Computation, RANDOM 2013, held in August 2013 in the USA. The total of 48 carefully reviewed and selected papers presented in this volume consist of 23 APPROX papers selected out of 46 submissions, and 25 RANDOM papers selected out of 52 submissions. APPROX 2013 focuses on algorithmic and complexity theoretic issues relevant to the development of efficient approximate solutions to computationally difficult problems, while RANDOM 2013 focuses on applications of randomness to computational and combinatorial problems.

Download or read book Pattern Recognition written by Katrin Franke and published by Springer Science & Business Media. This book was released on 2006-09-11 with total page 790 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book constitutes the refereed proceedings of the 28th Symposium of the German Association for Pattern Recognition, DAGM 2006. The book presents 32 revised full papers and 44 revised poster papers together with 5 invited papers. Topical sections include image filtering, restoration and segmentation, shape analysis and representation, recognition, categorization and detection, computer vision and image retrieval, machine learning and statistical data analysis, biomedical data analysis, and more.

Download or read book Experimental Algorithms written by Jan Vahrenhold and published by Springer Science & Business Media. This book was released on 2009-05-22 with total page 302 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book constitutes the refereed proceedings of the 8th International Symposium on Experimental and Efficient Algorithms, SEA 2009, held in Dortmund, Germany, in June 2009. The 23 revised full papers were carefully reviewed and selected from 64 submissions and present current research on experimental evaluation and engineering of algorithms, as well as in various aspects of computational optimization and its applications. Contributions are supported by experimental evaluation, methodological issues in the design and interpretation of experiments, the use of (meta-) heuristics, or application-driven case studies that deepen the understanding of a problem's complexity.

Download or read book Local Algorithms for Graph Partitioning and Finding Dense Subgraphs written by Reid Andersen and published by . This book was released on 2007 with total page 105 pages. Available in PDF, EPUB and Kindle. Book excerpt: This thesis is concerned with a new type of approximation algorithm for the fundamental problems of partitioning a graph and identifying its dense subgraphs. We develop local approximation algorithms for these two key problems, which search for a small set of vertices near a specified seed vertex, have a running time that is independent of the size of the graph, and for which we can prove a local approximation guarantee.

Download or read book On the Performance of Spectral Graph Partitioning Methods written by Stephen Guattery and published by . This book was released on 1994 with total page 29 pages. Available in PDF, EPUB and Kindle. Book excerpt: Abstract: "Computing graph separators is an important step in many graph algorithms. A popular technique for finding separators involves spectral methods. However, there is not much theoretical analysis of the quality of the separators produced by this technique; instead it is usually claimed that spectral methods 'work well in practice.' We present an initial attempt at such analysis. In particular, we consider two popular spectral separator algorithms, and provide counterexamples that show these algorithms perform poorly on certain graphs. We also consider a generalized definition of spectral methods that allows the use of some specified number of the eigenvectors corresponding to the smallest eigenvalues of the Laplacian matrix of a graph, and show that if such algorithms use a constant number of eigenvectors, then there are graphs for which they do no better than using only the second smallest eigenvector. Further, when applied to these graphs the algorithm based on the second smallest eigenvector performs poorly with respect to theoretical bounds. Even if an algorithm meeting the generalized definition uses up to n[superscript epsilon] for 0

Download or read book On the Performance of Spectral Graph Partitioning Methods written by Stephen Guattery and published by . This book was released on 1994 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: Abstract: "Computing graph separators is an important step in many graph algorithms. A popular technique for finding separators involves spectral methods. However, there is not much theoretical analysis of the quality of the separators produced by this technique; instead it is usually claimed that spectral methods 'work well in practice.' We present an initial attempt at such analysis. In particular, we consider two popular spectral separator algorithms, and provide counterexamples that show these algorithms perform poorly on certain graphs. We also consider a generalized definition of spectral methods that allows the use of some specified number of the eigenvectors corresponding to the smallest eigenvalues of the Laplacian matrix of a graph, and show that if such algorithms use a constant number of eigenvectors, then there are graphs for which they do no better than using only the second smallest eigenvector. Further, when applied to these graphs the algorithm based on the second smallest eigenvector performs poorly with respect to theoretical bounds. Even if an algorithm meeting the generalized definition uses up to n[superscript epsilon] for 0