Geometric Algorithms and Combinatorial Optimization

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 331-364

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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.

• 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

• 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

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