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Geometric Algorithms and Combinatorial Optimization

  • Martin Grötschel
  • László Lovász
  • Alexander Schrijver

Part of the Algorithms and Combinatorics book series (AC, volume 2)

Table of contents

  1. Front Matter
    Pages I-XII
  2. Martin Grötschel, László Lovász, Alexander Schrijver
    Pages 1-20
  3. Martin Grötschel, László Lovász, Alexander Schrijver
    Pages 21-45
  4. Martin Grötschel, László Lovász, Alexander Schrijver
    Pages 46-63
  5. Martin Grötschel, László Lovász, Alexander Schrijver
    Pages 64-101
  6. Martin Grötschel, László Lovász, Alexander Schrijver
    Pages 102-132
  7. Martin Grötschel, László Lovász, Alexander Schrijver
    Pages 133-156
  8. Martin Grötschel, László Lovász, Alexander Schrijver
    Pages 157-196
  9. Martin Grötschel, László Lovász, Alexander Schrijver
    Pages 197-224
  10. Martin Grötschel, László Lovász, Alexander Schrijver
    Pages 225-271
  11. Martin Grötschel, László Lovász, Alexander Schrijver
    Pages 272-303
  12. Martin Grötschel, László Lovász, Alexander Schrijver
    Pages 304-329
  13. Back Matter
    Pages 331-364

About this book

Introduction

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.

Keywords

Basis Reduction in Lattices Basisreduktion bei Gittern Combinatorics Convexity Ellipsoid Method Ellipsoidmethode Kombinatorische Optimierung Konvexität Lattice Linear Programming Lineares Programmieren Matching combinatorial optimization graph theory programming

Authors and affiliations

  • Martin Grötschel
    • 1
  • László Lovász
    • 2
  • Alexander Schrijver
    • 3
  1. 1.Institute of MathematicsUniversity of AugsburgAugsburgFed. Rep. of Germany
  2. 2.Department of Computer ScienceEötvös Loránd UniversityBudapestHungary
  3. 3.Department of EconometricsTilburg UniversityTilburgThe Netherlands

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-642-97881-4
  • Copyright Information Springer-Verlag Berlin Heidelberg 1988
  • Publisher Name Springer, Berlin, Heidelberg
  • eBook Packages Springer Book Archive
  • Print ISBN 978-3-642-97883-8
  • Online ISBN 978-3-642-97881-4
  • Series Print ISSN 0937-5511
  • Buy this book on publisher's site
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