Skip to main content
SpringerLink
Log in
Book cover

Geometric Algorithms and Combinatorial Optimization

  • Book
  • © 1988

Overview

Authors:
  1. Martin Grötschel

    1. Institute of Mathematics, University of Augsburg, Augsburg, Fed. Rep. of Germany

    View author publications

    You can also search for this author in PubMed   Google Scholar

  2. László Lovász

    1. Department of Computer Science, Eötvös Loránd University, Budapest, Hungary

    View author publications

    You can also search for this author in PubMed   Google Scholar

  3. Alexander Schrijver

    1. Department of Econometrics, Tilburg University, Tilburg, The Netherlands

    View author publications

    You can also search for this author in PubMed   Google Scholar

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

This is a preview of subscription content, log in via an institution to check access.

Access this book

Log in via an institution
eBook USD 74.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Other ways to access

Licence this eBook for your library

Institutional subscriptions

Table of contents (11 chapters)

  1. Front Matter

    Pages I-XII
    Download chapter PDF

  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 331-364
    Download chapter PDF
Back to top

Keywords

About this book

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.

Authors and Affiliations

  • Institute of Mathematics, University of Augsburg, Augsburg, Fed. Rep. of Germany

    Martin Grötschel

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

    László Lovász

  • Department of Econometrics, Tilburg University, Tilburg, The Netherlands

    Alexander Schrijver

Bibliographic Information

  • Book Title: Geometric Algorithms and Combinatorial Optimization

  • Authors: Martin Grötschel, László Lovász, Alexander Schrijver

  • Series Title: Algorithms and Combinatorics

  • DOI: https://doi.org/10.1007/978-3-642-97881-4

  • Publisher: Springer Berlin, Heidelberg

  • eBook Packages: Springer Book Archive

  • Copyright Information: Springer-Verlag Berlin Heidelberg 1988

  • eBook ISBN: 978-3-642-97881-4Published: 06 December 2012

  • Series ISSN: 0937-5511

  • Series E-ISSN: 2197-6783

  • Edition Number: 1

  • Number of Pages: XII, 362

  • Topics: Combinatorics

Publish with us

Policies and ethics

Back to top

Search

Navigation