基本説明
This most basic combinatorial optimization problem appears in a range of models in finance, logistics or general integer programming.
Full Description
Thirteen years have passed since the seminal book on knapsack problems by Martello and Toth appeared. On this occasion a former colleague exclaimed back in 1990: "How can you write 250 pages on the knapsack problem?" Indeed, the definition of the knapsack problem is easily understood even by a non-expert who will not suspect the presence of challenging research topics in this area at the first glance. However, in the last decade a large number of research publications contributed new results for the knapsack problem in all areas of interest such as exact algorithms, heuristics and approximation schemes. Moreover, the extension of the knapsack problem to higher dimensions both in the number of constraints and in the num ber of knapsacks, as well as the modification of the problem structure concerning the available item set and the objective function, leads to a number of interesting variations of practical relevance which were the subject of intensive research during the last few years. Hence, two years ago the idea arose to produce a new monograph covering not only the most recent developments of the standard knapsack problem, but also giving a comprehensive treatment of the whole knapsack family including the siblings such as the subset sum problem and the bounded and unbounded knapsack problem, and also more distant relatives such as multidimensional, multiple, multiple-choice and quadratic knapsack problems in dedicated chapters.
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.
