Geometric Algorithms and Combinatorial Optimization

Martin Grötschel, Laszlo Lovasz, Alexander Schrijver (Autoren)

Buch | Softcover

XII, 362 Seiten

2011 | 2nd ed. 1993. Softcover reprint of the original 2nd ed. 1993
Springer Berlin (Verlag)
978-3-642-78242-8 (ISBN)

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 Nevertheless, we do not feel that the ongoing research has made this book outdated.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

Prof. Dr. Dr. h.c. mult. Martin Grötschel, Technische Universität Berlin, Zuse-Institut Berlin (ZIB) und MATHEON.

László Lovász ist einer der Leiter der theoretischen Forschungsabteilung der Microsoft Corporation. Er hat 1999 den Wolf-Preis sowie den Gödel-Preis für die beste wissenschaftliche Veröffentlichung in der Informatik erhalten.

0. Mathematical Preliminaries.- 0.1 Linear Algebra and Linear Programming.- 0.2 Graph Theory.- 1. Complexity, Oracles, and Numerical Computation.- 1.1 Complexity Theory: P and NP.- 1.2 Oracles.- 1.3 Approximation and Computation of Numbers.- 1.4 Pivoting and Related Procedures.- 2. Algorithmic Aspects of Convex Sets: Formulation of the Problems.- 2.1 Basic Algorithmic Problems for Convex Sets.- 2.2 Nondeterministic Decision Problems for Convex Sets.- 3. The Ellipsoid Method.- 3.1 Geometric Background and an Informal Description.- 3.2 The Central-Cut Ellipsoid Method.- 3.3 The Shallow-Cut Ellipsoid Method.- 4. Algorithms for Convex Bodies.- 4.1 Summary of Results.- 4.2 Optimization from Separation.- 4.3 Optimization from Membership.- 4.4 Equivalence of the Basic Problems.- 4.5 Some Negative Results.- 4.6 Further Algorithmic Problems for Convex Bodies.- 4.7 Operations on Convex Bodies.- 5. Diophantine Approximation and Basis Reduction.- 5.1 Continued Fractions.- 5.2 Simultaneous Diophantine Approximation: Formulation of the Problems.- 5.3 Basis Reduction in Lattices.- 5.4 More on Lattice Algorithms.- 6. Rational Polyhedra.- 6.1 Optimization over Polyhedra: A Preview.- 6.2 Complexity of Rational Polyhedra.- 6.3 Weak and Strong Problems.- 6.4 Equivalence of Strong Optimization and Separation.- 6.5 Further Problems for Polyhedra.- 6.6 Strongly Polynomial Algorithms.- 6.7 Integer Programming in Bounded Dimension.- 7. Combinatorial Optimization: Some Basic Examples.- 7.1 Flows and Cuts.- 7.2 Arborescences.- 7.3 Matching.- 7.4 Edge Coloring.- 7.5 Matroids.- 7.6 Subset Sums.- 7.7 Concluding Remarks.- 8. Combinatorial Optimization: A Tour d'Horizon.- 8.1 Blocking Hypergraphs and Polyhedra.- 8.2 Problems on Bipartite Graphs.- 8.3 Flows, Paths, Chains, and Cuts.- 8.4 Trees,Branchings, and Rooted and Directed Cuts.- 8.5 Matchings, Odd Cuts, and Generalizations.- 8.6 Multicommodity Flows.- 9. Stable Sets in Graphs.- 9.1 Odd Circuit Constraints and t-Perfect Graphs.- 9.2 Clique Constraints and Perfect Graphs.- 9.3 Orthonormal Representations.- 9.4 Coloring Perfect Graphs.- 9.5 More Algorithmic Results on Stable Sets.- 10. Submodular Functions.- 10.1 Submodular Functions and Polymatroids.- 10.2 Algorithms for Polymatroids and Submodular Functions.- 10.3 Submodular Functions on Lattice, Intersecting, and Crossing Families.- 10.4 Odd Submodular Function Minimization and Extensions.- References.- Notation Index.- Author Index.

Erscheint lt. Verlag	21.12.2011
Reihe/Serie	Algorithms and Combinatorics
Zusatzinfo	XII, 362 p.
Verlagsort	Berlin
Sprache	englisch
Maße	170 x 242 mm
Gewicht	648 g
Themenwelt	Mathematik / Informatik ► Mathematik ► Graphentheorie
Schlagworte	Algorithmen • algorithms • Basis Reduction in Lattices • Basisreduktion bei Gittern • combinatorial optimization • combinatorics • Convexity • Ellipsoid Method • Ellipsoidmethode • Kombinatorische Optimierung • Konvexität • lattice • Lineares Programmieren • Linear Programming • operations resear
ISBN-10	3-642-78242-6 / 3642782426
ISBN-13	978-3-642-78242-8 / 9783642782428
Zustand	Neuware