This book provides a full-scale presentation of all methods and techniques available for the solution of the Knapsack problem. This most basic combinatorial optimization problem appears explicitly or as a subproblem in a wide range of optimization models with backgrounds such diverse as cutting and packing, finance, logistics or general integer programming. This monograph spans the range from a comprehensive introduction of classical algorithmic methods to the unified presentation of the most recent and advanced results in this area many of them originating from the authors. The chapters dealing with particular versions and extensions of the Knapsack problem are self-contained to a high degree and provide a valuable source of reference for researchers. Due to its simple structure, the Knapsack problem is an ideal model for introducing solution techniques to students of computer science, mathematics and economics. The first three chapters give an in-depth treatment of several basic techniques, making the book also suitable as underlying literature for courses in combinatorial optimization and approximation.

Table of Contents:
1 Introduction.- 1.1 Introducing the Knapsack Problem.- 1.2 Variants and Extensions of the Knapsack Pr©blem.- 1.3 Single-Capacity Versus All-Capacities Problem.- 1.4 Assumptions on the Input Data.- 1.5 Performance of Algorithms.- 2. Basic Algorithmic Concepts.- 2.1 The Greedy Algorithm.- 2.2 Linear Programming Relaxation.- 2.3 Dynamic Programming.- 2.4 Branch-and-Bound.- 2.5 Approximation Algorithms.- 2.6 Approximation Schemes.- 3. Advanced Algorithmic Concepts.- 3.1 Finding the Split Item in Linear Time.- 3.2 Variable Reduction.- 3.3 Storage Reduction in Dynamic Programming.- 3.4 Dynamic Programming with Lists.- 3.5 Combining Dynamic Programming and Upper Bounds.- 3.6 Balancing.- 3.7 Word RAM Algorithms.- 3.8 Relaxations.- 3.9 Lagrangian Decomposition.- 3.10 The Knapsack Polytope.- 4. The Subset Sum Problem.- 4.1 Dynamic Programming.- 4.2 Branch-and-Bound.- 4.3 Core Algorithms.- 4.4 Computational Results: Exact Algorithms.- 4.5 Polynomial Time Approximation Schemes for Subset Sum.- 4.6 A Fully Polynomial Time Approximation Scheme for Subset Sum.- 4.7 Computational Results: FPTAS.- 5. Exact Solution of the Knapsack Problem.- 5.1 Branch-and-Bound.- 5.2 Primal Dynamic Programming Algorithms.- 5.3 Primal-Dual Dynamic Programming Algorithms.- 5.4 The Core Concept.- 5.5 Computational Experiments.- 6. Approximation Algorithms for the Knapsack Problem.- 6.1 Polynomial Time Approximation Schemes.- 6.2 Fully Polynomial Time Approximation Schemes.- 7. The Bounded Knapsack Problem.- 7.1 Introduction.- 7.2 Dynamic Programming.- 7.3 Branch-and-Bound.- 7.4 Approximation Algorithms.- 8. The Unbounded Knapsack Problem.- 8.1 Introduction.- 8.2 Periodicity and Dominance.- 8.3 Dynamic Programming.- 8.4 Branch-and-Bound.- 8.5 Approximation Algorithms.- 9 Multidimensional KnapsackProblems.- 9.1 Introduction.- 9.2 Relaxations and Reductions.- 9.3 Exact Algorithms.- 9.4 Approximation.- 9.5 Heuristic Algorithms.- 9.6 The Two-Dimensional Knapsack Problem.- 9.7 The Cardinality Constrained Knapsack Problem.- 9.8 The Multidimensional Multiple-Choice Knapsack Problem.- 10. Multiple Knapsack Problems.- 10.1 Introduction.- 10.2 Upper Bounds.- 10.3 Branch-and-Bound.- 10.4 Approximation Algorithms.- 10.5 Polynomial Time Approximation Schemes.- 10.6 Variants of the Multiple Knapsack Problem.- 11. The Multiple-Choice Knapsack Problem.- 11.1 Introduction.- 11.2 Dominance and Upper Bounds.- 11.3 Class Reduction.- 11.4 Branch-and-Bound.- 11.5 Dynamic Programming.- 11.6 Reduction of States.- 11.7 Hybrid Algorithms and Expanding Core Algorithms.- 11.8 Computational Experiments.- 11.9 Heuristics and Approximation Algorithms.- 11.10 Variants of the Multiple-Choice Knapsack Problem.- 12. The Quadratic Knapsack Problem.- 12.1 Introduction.- 12.2 Upper Bounds.- 12.3 Variable Reduction.- 12.4 Branch-and-Bound.- 12.5 The Algorithm by Caprara, Pisinger and Toth.- 12.6 Heuristics.- 12.7 Approximation Algorithms.- 12.8 Computational Experiments Exact Algorithms.- 12.9 Computational Experiments Upper Bounds.- 13. Other Knapsack Problems.- 13.1 Multiobjective Knapsack Problems.- 13.2 The Precedence Constraint Knapsack Problem (PCKP).- 13.3 Further Variants.- 14. Stochastic Aspects of Knapsack Problems.- 14.1 The Probabilistic Model.- 14.2 Structural Results.- 14.3 Algorithms with Expected Performance Guarantee.- 14.4 Expected Performance of Greedy-Type Algorithms.- 14.5 Algorithms with Expected Running Time.- 14.6 Results for the Subset Sum Problem.- 14.7 Results for the Multidimensional Knapsack Problem.- 14.8 The On-Line Knapsack Problem.- 15. Some Selected Applications.-15.1 Two-Dimensional Two-Stage Cutting Problems.- 15.2 Column Generation in Cutting Stock Problems.- 15.3 Separation of Cover Inequalities.- 15.4 Financial Decision Problems.- 15.5 Asset-Backed Securitization.- 15.6 Knapsack Cryptosystems.- 15.7 Combinatorial Auctions.- A. Introduction to NP-Completeness of Knapsack Problems.- A.1 Definitions.- A.2 NP-Completeness of the Subset Sum Problem.- A.3 NP-Completeness of the Knapsack Problem.- A.4 NP-Completeness of Other Knapsack Problems.- References.- Author Index.