Riassunto
This book develops geometric techniques for proving the polynomial time solvability of problems in convexity theory, geometry, and, in particular, combinatorial optimization. The book is a continuation and extension of previous research of the authors for which they received the Fulkerson prize, awarded by the Mathematical Programming Society and the American Mathematical Society. To quote from a review: "This book ... is doubtless one of the outstanding books in discrete mathematics at all." (Journal of Information Proceedings and Cybernetics).
Contenuti
0. Mathematical Preliminaries.- 0.1 Linear Algebra and Linear Programming.- Basic Notation.- Hulls, Independence, Dimension.- Eigenvalues, Positive Definite Matrices.- Vector Norms, Balls.- Matrix Norms.- Some Inequalities.- Polyhedra, Inequality Systems.- Linear (Diophantine) Equations and Inequalities.- Linear Programming and Duality.- 0.2 Graph Theory.- Graphs.- Digraphs.- Walks, Paths, Circuits, Trees.- 1. Complexity, Oracles, and Numerical Computation.- 1.1 Complexity Theory: P and NP.- Problems.- Algorithms and Turing Machines.- Encoding.- Time and Space Complexity.- Decision Problems: The Classes P and NP.- 1.2 Oracles.- The Running Time of Oracle Algorithms.- Transformation and Reduction.- NP-Completeness and Related Notion.- 1.3 Approximation and Computation of Numbers.- Encoding Length of Numbers.- Polynomial and Strongly Polynomial Computations.- Polynomial Time Approximation of Real Numbers.- 1.4 Pivoting and Related Procedures.- Gaussian Elimination.- Gram-Schmidt Orthogonalization.- The Simplex Method.- Computation of the Hermite Normal Form.- 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.- Properties of Ellipsoids.- Description of the Basic Ellipsoid Method.- Proofs of Some Lemmas.- Implementation Problems and Polynomiality.- Some Examples.- 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.- The Sum.- The Convex Hull of the Union.- The Intersection.- Polars, Blockers, Antiblockers.- 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.- Arborescences and Rooted Cuts.- Trees and Cuts in Undirected Graphs.- Dicuts and Dijoins.- 8.5 Matchings, Odd Cuts, and Generalizations.- Matching.- b-Matching.- T-Joins and T-Cuts.- Chinese Postmen and Traveling Salesmen.- 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.- Antiblockers of Hypergraphs.- 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.- Packing Bases of a Matroid.- 10.3 Submodular Functions on Lattice, Intersecting, and Crossing Families.- 10.4 Odd Submodular Function Minimization and Extensions.- References.- Notation Index.- Author Index.
Le informazioni nella sezione "Su questo libro" possono far riferimento a edizioni diverse di questo titolo.
