Hans Kellerer, Ulrich Pferschy, David Pisinger

Knapsack Problems

• Gebundenes Buch

Produktbeschreibung

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.

---

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.

Produktdetails

• Verlag: Springer / Springer Berlin Heidelberg / Springer, Berlin
• Artikelnr. des Verlages: 978-3-540-40286-2
• 2004
• Seitenzahl: 572
• Erscheinungstermin: 15. Dezember 2003
• Englisch
• Abmessung: 241mm x 160mm x 36mm
• Gewicht: 964g
• ISBN-13: 9783540402862
• ISBN-10: 3540402861
• Artikelnr.: 12352197

Autorenporträt

Hans Kellerer, University of Graz, Austria / Ulrich Pferschy, University of Graz, Austria / David Pisinger, University of Copenhagen, Denmark

Inhaltsangabe

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.

Rezensionen

From the reviews: "The book explores the knapsack problem and its variants in 15 chapters ... . On the whole, the authors present a rich amount of material, much of which belongs to the most recent advancement in the subject ... . This self-contained monograph is a valuable addition to the existing literature on knapsack problems. It will certainly make an excellent reference for researchers in combinatorial optimization. ... In this regard, the libraries in this field-should have a copy of this book on their shelves." (J Xue, Journal of the Operational Research Society, Vol. 56 (11), 2005) "This book provides a comprehensive overview of the methods for solving KP, its variants and generalizations. ... By presenting a range of algorithmic techniques, this book is also a suitable tool for studying modern algorithms. ... With its more than 500 references and its author and subject indexes, this book will be a valuable tool for many researchers in combinatorial optimization." (Peter Butkovic, Mathematical Reviews, 2006 d) "This book presents a large number of new results for knapsack problems, especially related to exact and approximate (heuristic) algorithms. Moreover, many modifications and extensions of the knapsack problem are treated. To do this, the authors found and consequently used a close connection between manifold contents and a clear, stimulating presentation. ... With an introduction into NP-completeness of knapsack problems a monograph ends, which spans the range from a comprehensive introduction to the most recent and advanced results very nicely." (Jürgen Köhler, OR Spectrum, Issue 27, 2005) "The book starts with a basic introduction to the knapsack problem ... . It proceeds with a discussion about the basic techniques available for the solution to this problem, which makes it useful reading for undergraduate students of computer science, mathematics and economics. ... In conclusion, it could be said that the book is a valuable source for students, but also for researchers interested in this problem." (K. soric, Zentralblatt MATH, Vol. 1103 (5), 2007)