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Geometric Algorithms and Combinatorial Optimization

  • Book
  • © 1993

Overview

Authors:
  1. Martin Grötschel

    1. Konrad-Zuse-Zentrum für Informationstechnik Berlin, Berlin, Germany
      Fachbereich Mathematik, Technische Universität Berlin, Berlin, Germany

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  2. László Lovász

    1. Department of Computer Science, Eötvös Loránd University, Budapest, Hungary
      Department of Mathematics, Yale University, New Haven, USA

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  3. Alexander Schrijver

    1. CWI (Center for Mathematics and Computer Science), Amsterdam, The Netherlands
      Department of Mathematics, University of Amsterdam, Amsterdam, The Netherlands

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

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Table of contents (11 chapters)

  1. Front Matter

    Pages I-XII
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  2. Mathematical Preliminaries

    • Martin Grötschel, László Lovász, Alexander Schrijver
    Pages 1-20

  3. Complexity, Oracles, and Numerical Computation

    • Martin Grötschel, László Lovász, Alexander Schrijver
    Pages 21-45

  4. Algorithmic Aspects of Convex Sets: Formulation of the Problems

    • Martin Grötschel, László Lovász, Alexander Schrijver
    Pages 46-63

  5. The Ellipsoid Method

    • Martin Grötschel, László Lovász, Alexander Schrijver
    Pages 64-101

  6. Algorithms for Convex Bodies

    • Martin Grötschel, László Lovász, Alexander Schrijver
    Pages 102-132

  7. Diophantine Approximation and Basis Reduction

    • Martin Grötschel, László Lovász, Alexander Schrijver
    Pages 133-156

  8. Rational Polyhedra

    • Martin Grötschel, László Lovász, Alexander Schrijver
    Pages 157-196

  9. Combinatorial Optimization: Some Basic Examples

    • Martin Grötschel, László Lovász, Alexander Schrijver
    Pages 197-224

  10. Combinatorial Optimization: A Tour d’Horizon

    • Martin Grötschel, László Lovász, Alexander Schrijver
    Pages 225-271

  11. Stable Sets in Graphs

    • Martin Grötschel, László Lovász, Alexander Schrijver
    Pages 272-303

  12. Submodular Functions

    • Martin Grötschel, László Lovász, Alexander Schrijver
    Pages 304-329

  13. Back Matter

    Pages 331-363
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Keywords

About this book

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.

Authors and Affiliations

  • Konrad-Zuse-Zentrum für Informationstechnik Berlin, Berlin, Germany

    Martin Grötschel

  • Fachbereich Mathematik, Technische Universität Berlin, Berlin, Germany

    Martin Grötschel

  • Department of Computer Science, Eötvös Loránd University, Budapest, Hungary

    László Lovász

  • Department of Mathematics, Yale University, New Haven, USA

    László Lovász

  • CWI (Center for Mathematics and Computer Science), Amsterdam, The Netherlands

    Alexander Schrijver

  • Department of Mathematics, University of Amsterdam, Amsterdam, The Netherlands

    Alexander Schrijver

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