Geometric Algorithms And Combinatorial Optimization

Author : Martin Grötschel,Laszlo Lovasz,Alexander Schrijver
Publisher : Springer Science & Business Media
Page : 374 pages
File Size : 43,5 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9783642978814

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Geometric Algorithms and Combinatorial Optimization by Martin Grötschel,Laszlo Lovasz,Alexander Schrijver Pdf

Historically, there is a close connection between geometry and optImization. This is illustrated by methods like the gradient method and the simplex method, which are associated with clear geometric pictures. In combinatorial optimization, however, many of the strongest and most frequently used algorithms are based on the discrete structure of the problems: the greedy algorithm, shortest path and alternating path methods, branch-and-bound, etc. In the last several years geometric methods, in particular polyhedral combinatorics, have played a more and more profound role in combinatorial optimization as well. Our book discusses two recent geometric algorithms that have turned out to have particularly interesting consequences in combinatorial optimization, at least from a theoretical point of view. These algorithms are able to utilize the rich body of results in polyhedral combinatorics. The first of these algorithms is the ellipsoid method, developed for nonlinear programming by N. Z. Shor, D. B. Yudin, and A. S. NemirovskiI. It was a great surprise when L. G. Khachiyan showed that this method can be adapted to solve linear programs in polynomial time, thus solving an important open theoretical problem. While the ellipsoid method has not proved to be competitive with the simplex method in practice, it does have some features which make it particularly suited for the purposes of combinatorial optimization. The second algorithm we discuss finds its roots in the classical "geometry of numbers", developed by Minkowski. This method has had traditionally deep applications in number theory, in particular in diophantine approximation.

Author : Vladimir Boltyanski,Horst Martini,V. Soltan
Publisher : Springer Science & Business Media
Page : 438 pages
File Size : 44,8 Mb
Release : 2013-12-11
Category : Mathematics
ISBN : 9781461553199

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Geometric Methods and Optimization Problems by Vladimir Boltyanski,Horst Martini,V. Soltan Pdf

VII Preface In many fields of mathematics, geometry has established itself as a fruitful method and common language for describing basic phenomena and problems as well as suggesting ways of solutions. Especially in pure mathematics this is ob vious and well-known (examples are the much discussed interplay between lin ear algebra and analytical geometry and several problems in multidimensional analysis). On the other hand, many specialists from applied mathematics seem to prefer more formal analytical and numerical methods and representations. Nevertheless, very often the internal development of disciplines from applied mathematics led to geometric models, and occasionally breakthroughs were b~ed on geometric insights. An excellent example is the Klee-Minty cube, solving a problem of linear programming by transforming it into a geomet ric problem. Also the development of convex programming in recent decades demonstrated the power of methods that evolved within the field of convex geometry. The present book focuses on three applied disciplines: control theory, location science and computational geometry. It is our aim to demonstrate how methods and topics from convex geometry in a wider sense (separation theory of convex cones, Minkowski geometry, convex partitionings, etc.) can help to solve various problems from these disciplines.

Author : Herbert Edelsbrunner
Publisher : Springer Science & Business Media
Page : 423 pages
File Size : 52,9 Mb
Release : 2012-12-06
Category : Computers
ISBN : 9783642615689

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Algorithms in Combinatorial Geometry by Herbert Edelsbrunner Pdf

Computational geometry as an area of research in its own right emerged in the early seventies of this century. Right from the beginning, it was obvious that strong connections of various kinds exist to questions studied in the considerably older field of combinatorial geometry. For example, the combinatorial structure of a geometric problem usually decides which algorithmic method solves the problem most efficiently. Furthermore, the analysis of an algorithm often requires a great deal of combinatorial knowledge. As it turns out, however, the connection between the two research areas commonly referred to as computa tional geometry and combinatorial geometry is not as lop-sided as it appears. Indeed, the interest in computational issues in geometry gives a new and con structive direction to the combinatorial study of geometry. It is the intention of this book to demonstrate that computational and com binatorial investigations in geometry are doomed to profit from each other. To reach this goal, I designed this book to consist of three parts, acorn binatorial part, a computational part, and one that presents applications of the results of the first two parts. The choice of the topics covered in this book was guided by my attempt to describe the most fundamental algorithms in computational geometry that have an interesting combinatorial structure. In this early stage geometric transforms played an important role as they reveal connections between seemingly unrelated problems and thus help to structure the field.

Author : Michel Marie Deza,Monique Laurent
Publisher : Springer
Page : 580 pages
File Size : 48,5 Mb
Release : 2009-11-12
Category : Mathematics
ISBN : 9783642042959

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Geometry of Cuts and Metrics by Michel Marie Deza,Monique Laurent Pdf

Cuts and metrics are well-known objects that arise - independently, but with many deep and fascinating connections - in diverse fields: in graph theory, combinatorial optimization, geometry of numbers, combinatorial matrix theory, statistical physics, VLSI design etc. This book presents a wealth of results, from different mathematical disciplines, in a unified comprehensive manner, and establishes new and old links, which cannot be found elsewhere. It provides a unique and invaluable source for researchers and graduate students. From the Reviews: "This book is definitely a milestone in the literature of integer programming and combinatorial optimization. It draws from the Interdisciplinarity of these fields [...]. With knowledge about the relevant terms, one can enjoy special subsections without being entirely familiar with the rest of the chapter. This makes it not only an interesting research book but even a dictionary. [...] The longer one works with it, the more beautiful it becomes." Optima 56, 1997.

Author : Martin Grötschel,Laszlo Lovasz,Alexander Schrijver
Publisher : Springer Science & Business Media
Page : 374 pages
File Size : 41,6 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9783642782404

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Geometric Algorithms and Combinatorial Optimization by Martin Grötschel,Laszlo Lovasz,Alexander Schrijver Pdf

Since the publication of the first edition of our book, geometric algorithms and combinatorial optimization have kept growing at the same fast pace as before. Nevertheless, we do not feel that the ongoing research has made this book outdated. Rather, it seems that many of the new results build on the models, algorithms, and theorems presented here. For instance, the celebrated Dyer-Frieze-Kannan algorithm for approximating the volume of a convex body is based on the oracle model of convex bodies and uses the ellipsoid method as a preprocessing technique. The polynomial time equivalence of optimization, separation, and membership has become a commonly employed tool in the study of the complexity of combinatorial optimization problems and in the newly developing field of computational convexity. Implementations of the basis reduction algorithm can be found in various computer algebra software systems. On the other hand, several of the open problems discussed in the first edition are still unsolved. For example, there are still no combinatorial polynomial time algorithms known for minimizing a submodular function or finding a maximum clique in a perfect graph. Moreover, despite the success of the interior point methods for the solution of explicitly given linear programs there is still no method known that solves implicitly given linear programs, such as those described in this book, and that is both practically and theoretically efficient. In particular, it is not known how to adapt interior point methods to such linear programs.

Author : Christos H. Papadimitriou,Kenneth Steiglitz
Publisher : Courier Corporation
Page : 528 pages
File Size : 54,8 Mb
Release : 2013-04-26
Category : Mathematics
ISBN : 9780486320137

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Combinatorial Optimization by Christos H. Papadimitriou,Kenneth Steiglitz Pdf

This graduate-level text considers the Soviet ellipsoid algorithm for linear programming; efficient algorithms for network flow, matching, spanning trees, and matroids; the theory of NP-complete problems; local search heuristics for NP-complete problems, more. 1982 edition.

Author : Alexander Schrijver
Publisher : Springer Science & Business Media
Page : 2024 pages
File Size : 41,8 Mb
Release : 2003-02-12
Category : Business & Economics
ISBN : 3540443894

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Combinatorial Optimization by Alexander Schrijver Pdf

From the reviews: "About 30 years ago, when I was a student, the first book on combinatorial optimization came out referred to as "the Lawler" simply. I think that now, with this volume Springer has landed a coup: "The Schrijver". The box is offered for less than 90.- EURO, which to my opinion is one of the best deals after the introduction of this currency." OR-Spectrum

Author : Herbert Edelsbrunner
Publisher : Springer Science & Business Media
Page : 446 pages
File Size : 54,9 Mb
Release : 1987-07-31
Category : Computers
ISBN : 354013722X

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Algorithms in Combinatorial Geometry by Herbert Edelsbrunner Pdf

Computational geometry as an area of research in its own right emerged in the early seventies of this century. Right from the beginning, it was obvious that strong connections of various kinds exist to questions studied in the considerably older field of combinatorial geometry. For example, the combinatorial structure of a geometric problem usually decides which algorithmic method solves the problem most efficiently. Furthermore, the analysis of an algorithm often requires a great deal of combinatorial knowledge. As it turns out, however, the connection between the two research areas commonly referred to as computa tional geometry and combinatorial geometry is not as lop-sided as it appears. Indeed, the interest in computational issues in geometry gives a new and con structive direction to the combinatorial study of geometry. It is the intention of this book to demonstrate that computational and com binatorial investigations in geometry are doomed to profit from each other. To reach this goal, I designed this book to consist of three parts, acorn binatorial part, a computational part, and one that presents applications of the results of the first two parts. The choice of the topics covered in this book was guided by my attempt to describe the most fundamental algorithms in computational geometry that have an interesting combinatorial structure. In this early stage geometric transforms played an important role as they reveal connections between seemingly unrelated problems and thus help to structure the field.

Author : Daniel Scholz
Publisher : Springer Science & Business Media
Page : 142 pages
File Size : 47,9 Mb
Release : 2011-11-06
Category : Mathematics
ISBN : 9781461419518

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Deterministic Global Optimization by Daniel Scholz Pdf

This monograph deals with a general class of solution approaches in deterministic global optimization, namely the geometric branch-and-bound methods which are popular algorithms, for instance, in Lipschitzian optimization, d.c. programming, and interval analysis.It also introduces a new concept for the rate of convergence and analyzes several bounding operations reported in the literature, from the theoretical as well as from the empirical point of view. Furthermore, extensions of the prototype algorithm for multicriteria global optimization problems as well as mixed combinatorial optimization problems are considered. Numerical examples based on facility location problems support the theory. Applications of geometric branch-and-bound methods, namely the circle detection problem in image processing, the integrated scheduling and location makespan problem, and the median line location problem in the three-dimensional space are also presented. The book is intended for both researchers and students in the areas of mathematics, operations research, engineering, and computer science.

Author : Andreas S. Schulz,Martin Skutella,Sebastian Stiller,Dorothea Wagner
Publisher : Springer
Page : 150 pages
File Size : 41,5 Mb
Release : 2016-01-31
Category : Business & Economics
ISBN : 9783319249711

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Gems of Combinatorial Optimization and Graph Algorithms by Andreas S. Schulz,Martin Skutella,Sebastian Stiller,Dorothea Wagner Pdf

Are you looking for new lectures for your course on algorithms, combinatorial optimization, or algorithmic game theory? Maybe you need a convenient source of relevant, current topics for a graduate student or advanced undergraduate student seminar? Or perhaps you just want an enjoyable look at some beautiful mathematical and algorithmic results, ideas, proofs, concepts, and techniques in discrete mathematics and theoretical computer science? Gems of Combinatorial Optimization and Graph Algorithms is a handpicked collection of up-to-date articles, carefully prepared by a select group of international experts, who have contributed some of their most mathematically or algorithmically elegant ideas. Topics include longest tours and Steiner trees in geometric spaces, cartograms, resource buying games, congestion games, selfish routing, revenue equivalence and shortest paths, scheduling, linear structures in graphs, contraction hierarchies, budgeted matching problems, and motifs in networks. This volume is aimed at readers with some familiarity of combinatorial optimization, and appeals to researchers, graduate students, and advanced undergraduate students alike.

Author : David Avis,Alain Hertz,Odile Marcotte
Publisher : Springer Science & Business Media
Page : 273 pages
File Size : 55,6 Mb
Release : 2005-12-06
Category : Business & Economics
ISBN : 9780387255927

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Graph Theory and Combinatorial Optimization by David Avis,Alain Hertz,Odile Marcotte Pdf

Graph theory is very much tied to the geometric properties of optimization and combinatorial optimization. Moreover, graph theory's geometric properties are at the core of many research interests in operations research and applied mathematics. Its techniques have been used in solving many classical problems including maximum flow problems, independent set problems, and the traveling salesman problem. Graph Theory and Combinatorial Optimization explores the field's classical foundations and its developing theories, ideas and applications to new problems. The book examines the geometric properties of graph theory and its widening uses in combinatorial optimization theory and application. The field's leading researchers have contributed chapters in their areas of expertise.

Author : Levent Tuncel
Publisher : American Mathematical Soc.
Page : 233 pages
File Size : 53,5 Mb
Release : 2024-06-01
Category : Mathematics
ISBN : 9780821871850

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Polyhedral and Semidefinite Programming Methods in Combinatorial Optimization by Levent Tuncel Pdf

Author : Bernhard Korte,Jens Vygen
Publisher : Springer Science & Business Media
Page : 596 pages
File Size : 50,9 Mb
Release : 2006-01-27
Category : Mathematics
ISBN : 9783540292975

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Combinatorial Optimization by Bernhard Korte,Jens Vygen Pdf

This well-written textbook on combinatorial optimization puts special emphasis on theoretical results and algorithms with provably good performance, in contrast to heuristics. The book contains complete (but concise) proofs, as well as many deep results, some of which have not appeared in any previous books.

Author : Charles J. Colbourn,Roberto Grossi,Nadia Pisanti
Publisher : Springer
Page : 456 pages
File Size : 43,5 Mb
Release : 2019-07-11
Category : Computers
ISBN : 3030250040

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Combinatorial Algorithms by Charles J. Colbourn,Roberto Grossi,Nadia Pisanti Pdf

This book constitutes the refereed post-conference proceedings of the 30th International Workshop on Combinatorial Algorithms, IWOCA 2019, held in Pisa, Italy, in July 2019. The 36 regular papers presented in this volume were carefully reviewed and selected from 73 submissions. They cover diverse areas of combinatorical algorithms, complexity theory, graph theory and combinatorics, combinatorial optimization, cryptography and information security, algorithms on strings and graphs, graph drawing and labelling, computational algebra and geometry, computational biology, probabilistic and randomized algorithms, algorithms for big data analytics, and new paradigms of computation.

Author : Ding-Zhu Du,Panos M. Pardalos
Publisher : Springer Science & Business Media
Page : 395 pages
File Size : 51,5 Mb
Release : 2006-08-18
Category : Business & Economics
ISBN : 9780387238302

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Handbook of Combinatorial Optimization by Ding-Zhu Du,Panos M. Pardalos Pdf

This is a supplementary volume to the major three-volume Handbook of Combinatorial Optimization set. It can also be regarded as a stand-alone volume presenting chapters dealing with various aspects of the subject in a self-contained way.