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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 331-363

About this book

Introduction

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 Nevertheless, we do not feel that the ongoing research has made this book outdated.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.

Keywords

Basis Reduction in Lattices Basisreduktion bei Gittern Convexity Ellipsoid Method Ellipsoidmethode Kombinatorische Optimierung Konvexität Lattice Linear Programming Lineares Programmieren algorithms combinatorial optimization operations resear

Authors and affiliations

  • Martin Grötschel
    • 1
    • 2
  • László Lovász
    • 3
    • 4
  • Alexander Schrijver
    • 5
    • 6
  1. 1.Konrad-Zuse-Zentrum für Informationstechnik BerlinBerlinGermany
  2. 2.Fachbereich MathematikTechnische Universität BerlinBerlinGermany
  3. 3.Department of Computer ScienceEötvös Loránd UniversityBudapestHungary
  4. 4.Department of MathematicsYale UniversityNew HavenUSA
  5. 5.CWI (Center for Mathematics and Computer Science)AmsterdamThe Netherlands
  6. 6.Department of MathematicsUniversity of AmsterdamAmsterdamThe Netherlands

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-642-78240-4
  • Copyright Information Springer-Verlag Berlin Heidelberg 1993
  • Publisher Name Springer, Berlin, Heidelberg
  • eBook Packages Springer Book Archive
  • Print ISBN 978-3-642-78242-8
  • Online ISBN 978-3-642-78240-4
  • Series Print ISSN 0937-5511
  • Buy this book on publisher's site