Integer and Combinatorial Optimization

Transcription

1 Integer and Combinatorial Optimization GEORGE NEMHAUSER School of Industrial and Systems Engineering Georgia Institute of Technology Atlanta, Georgia LAURENCE WOLSEY Center for Operations Research and Econometrics Universite Catholique de Louvain Louvain-la-Neuve, Belgium A Wiley-Interscience Publication JOHN WILEY & SONS, INC. New York Chichester Weinheim Brisbane Singapore Toronto

2 PARTI. FOUNDATIONS The Scope of Integer and Combinatorial Optimization 3 1. Introduction 3 2. Modeling with Binary Variables I: Knapsack, Assignment and Matching, Covering, Packing and Partitioning 5 3. Modeling with Binary Variables II: Facility Location, Fixed-Charge Network Flow, and Traveling Salesman 7 4. Modeling with Binary Variables HI: Nonlinear Functions and Disjunctive Constraints Choices in Model Formulation Preprocessing Notes Exercises Linear Programming Introduction Duality The Primal and Dual Simplex Algorithms Subgradient Optimization Notes Graphs and Networks Introduction * The Minimum-Weight or Shortest-Path Problem The Minimum-Weight Spanning Tree Problem The Maximum-Flow and Minimum-Cut Problems The Transportation Problem: A Primal-Dual Algorithm A Primal Simplex Algorithm for Network Flow Problems Notes Polyhedral Theory Introduction and Elementary Linear Algebra Definitions ofpolyhedra and Dimension Describing Polyhedra by Facets Describing Polyhedra by Extreme Points and Extreme Rays Polarity 98 xi

3 xii 6. Polyhedral Ties Between Linear and Integer Programs Notes Exercises Computational Complexity Introduction Measuring Algorithm Efficiency and Problem Complexity Some Problems Solvable in Polynomial Time Remarks on 0-1 and Pure-Integer Programming Nondeterministic Polynomial-Time Algorithms and Jf& Problems The Most Difficult Jf& Problems: The Class Jf9* Complexity and Polyhedra Notes Exercises Polynomial-Time Algorithms for Linear Programming Introduction The Ellipsoid Algorithm The Polynomial Equivalence of Separation and Optimization A Projective Algorithm A Strongly Polynomial Algorithm for Combinatorial Linear Programs Notes Integer Lattices Introduction The Euclidean Algorithm Continued Fractions Lattices and Hermite Normal Form Reduced Bases Notes Exercises 202 PART II. GENERAL INTEGER PROGRAMMING 203 II.l The Theory of Valid Inequalities Introduction Generating All Valid Inequalities Gomory's Fractional Cuts and Rounding Superadditive Functions and Valid Inequalities A Polyhedral Description of Superadditive Valid Inequalitiesrfor Independence Systems Valid Inequalities for Mixed-Integer Sets Superadditivity for Mixed-Integer Sets Notes Exercises 256

4 xiii 11.2 Strong Valid Inequalities and Facets for Structured Integer Programs Introduction Valid Inequalities for the 0-1 Knapsack Polytope Valid Inequalities for the Symmetric Traveling Salesman Polytope Valid Inequalities for Variable Upper-Bound Flow Models Notes Exercises Duality and Relaxation Introduction Duality and the Value Function Superadditive Duality The Maximum-Weight Path Formulation and Superadditive Duality Modular Arithmetic and the Group Problem Lagrangian Relaxation and Duality Benders' Reformulation Notes Exercises Genera] Algorithms Introduction Branch-and-Bound Using Linear Programming Relaxations General Cutting-Plane Algorithms Notes Exercises 381 II.5 Special-Purpose Algorithms Introduction A Cutting-Plane Algorithm Using Strong Valid Inequalities Primal and Dual Heuristic Algorithms Decomposition Algorithms Dynamic Programming > Notes * Exercises 427 II.6 Applications of Special-Purpose Algorithms Knapsack and Group Problems Integer Programming Problems " The Symmetric Traveling Salesman Problem Fixed-Charge Network Flow Problems Applications of Basis Reduction Notes Exercises 526

5 xiv PART III. COMBINATORIAL OPTIMIZATION Integral Polyhedra Introduction Totally Unimodular Matrices Network Matrices Balanced and Totally Balanced Matrices Node Packing and Perfect Graphs Blocking and Integral Polyhedra Notes Exercises Matching Introduction Maximum-Cardinality Matching Maximum-Weight Matching Additional Results on Matching and Related Problems Notes Exercises Matroid and Submodular Function Optimization Introduction Elementary Properties Maximum-Weight Independent Sets Matroid Intersection Weighted Matroid Intersection Polymatroids, Separation, and Submodular Function Minimization Algorithms To Minimize a Submodular Function Covering with Independent Sets and Matroid Partition Submodular Function Maximization Notes Exercises 714 References 721 Author Index 749 Subject Index * 755