A First Course in Combinatorial Optimization

• Main Author: Lee, Jon
• Format: eBook Book
• Language: English
• Published: Cambridge Cambridge University Press 2004
• Edition: 1
• Series: Cambridge Texts in Applied Mathematics
• Subjects: Combinatorial analysis; Combinatorial optimization
• ISBN: 0521010128; 0521811511; 9780521010122; 9780521811514
• DOI: 10.1017/CBO9780511616655

A First Course in Combinatorial Optimization is a 2004 text for a one-semester introductory graduate-level course for students of operations research, mathematics, and computer science. It is a self-contained treatment of the subject, requiring only some mathematical maturity. Topics include: linear...

Table of Contents:

• Cover -- Half-title -- Title -- Copyright -- Contents -- Preface -- Introduction -- 0 Polytopes and Linear Programming -- 0.1 Finite Systems of Linear Inequalities -- 0.2 Linear-Programming Duality -- 0.3 Basic Solutions and the Primal Simplex Method -- 0.4 Sensitivity Analysis -- 0.5 Polytopes -- 0.6 Lagrangian Relaxation -- 0.7 The Dual Simplex Method -- 0.8 Totally Unimodular Matrices, Graphs, and Digraphs -- 0.9 Further Study -- 1 Matroids and the Greedy Algorithm -- 1.1 Independence Axioms and Examples of Matroids -- 1.2 Circuit Properties -- 1.3 Representations -- 1.4 The Greedy Algorithm -- 1.5 Rank Properties -- 1.6 Duality -- 1.7 The Matroid Polytope -- 1.8 Further Study -- 2 Minimum-Weight Dipaths -- 2.1 No Negative-Weight Cycles -- 2.2 All-Pairs Minimum-Weight Dipaths -- 2.3 Nonnegative Weights -- 2.4 No Dicycles and Knapsack Programs -- 2.5 Further Study -- 3 Matroid Intersection -- 3.1 Applications -- 3.2 An Efficient Cardinality Matroid-Intersection Algorithm and Consequences -- 3.3 An Efficient Maximum-Weight Matroid-Intersection Algorithm -- 3.4 The Matroid-Intersection Polytope -- 3.5 Further Study -- 4 Matching -- 4.1 Augmenting Paths and Matroids -- 4.2 The Matching Polytope -- 4.3 Duality and a Maximum-Cardinality Matching Algorithm -- 4.4 Kuhn's Algorithm for the Assignment Problem -- 4.5 Applications of Weighted Matching -- 4.6 Further Study -- 5 Flows and Cuts -- 5.1 Source-Sink Flows and Cuts -- 5.2 An Efficient Maximum-Flow Algorithm and Consequences -- 5.3 Undirected Cuts -- 5.4 Further Study -- 6 Cutting Planes -- 6.1 Generic Cutting-Plane Method -- 6.2 Chvátal-Gomory Cutting Planes -- 6.3 Gomory Cutting Planes -- 6.4 Tightening a Constraint -- 6.5 Constraint Generation for Combinatorial-Optimization Problems -- 6.6 Further Study -- 7 Branch-&amp -- -Bound -- 7.1 Branch-&amp -- -Bound Using Linear-Programming Relaxation
• 7.2 Knapsack Programs and Group Relaxation -- 7.3 Branch-&amp -- -Bound for Optimal-Weight Hamiltonian Tour -- 7.4 Maximum-Entropy Sampling and Branch-&amp -- -Bound -- 7.5 Further Study -- 8 Optimizing Submodular Functions -- 8.1 Minimizing Submodular Functions -- 8.2 Minimizing Submodular Functions Over Odd Sets -- 8.3 Maximizing Submodular Functions -- 8.4 Further Study -- Appendix: Notation and Terminology -- A.1 Sets -- A.2 Algebra -- A.3 Graphs -- A.4 Digraphs -- References -- Background Reading -- Further Reading -- Indexes