Knapsack Problems

Abstract

PrefaceThirteen 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 number 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.Furthermore, attention is paid to a number of less frequently considered variants of the knapsack problem and to the study of stochastic aspects of the problem. To illustrate the high practical relevance of the knapsack family for many industrial and economic problems, a number of applications are described in more detail. They are selected subjectively from the innumerable occurrences of knapsack problems reported in the literature.Our above-mentioned colleague will be surprised to notice that even on the more than 500 pages of this book not all relevant topics could be treated in equal depth but decisions had to be made on where to go into details of constructions and proofs and where to concentrate on stating results and refer to the appropriate publications. Moreover, an editorial deadline had to be drawn at some point. In our case, we stopped looking for new publications at the end of June 2003. VI PrefaceThe audience we envision for this book is threefold: The first two chapters offer a very basic introduction to the knapsack problem and the main algorithmic concepts to derive optimal and approximate solution. Chapter 3 presents a number of advanced algorithmic techniques which are used throughout the later chapters of the book. The style of presentation in these three chapters is kept rather simple and assumes only minimal prerequisites. They should be accessible to students and graduates of business administration, economics and engineering as well as practitioners with little knowledge of algorithms and optimization.This first part of the book is also well suited to introd...

Recoverable robust knapsacks: the discrete scenario case

“…Different branch-and-cut algorithms are used to solve this problem in practice. A detailed introduction to the knapsack problem and its variations can be found in Martello and Toth [19] and Kellerer, Pfeschy and Pisinger [17].…”

Performance Analysis in Robust Optimization

“…Constraint (6) ensures that the budget capacity B is not exceeded. Being NP-complete, the computational complexity of this problem is harder than for the assignment problem (for a general survey on the knapsack problem, see [22]). …”

A Fully Polynomial-Time Approximation Scheme for Single-Item Stochastic Inventory Control with Discrete Demand

“…Regarding Condition 1, we would have liked to extend our framework to deal with multi-variate DP, i.e., to allow fixeddimensional state and action spaces. Unfortunately, this is unlikely to be successful, since it is known that the existence of an FPTAS for the 2-dimensional 0/1 knapsack problem (which can be formulated as a 2-dimensional nondecreasing DP) would imply P = N P (see p. 252 in [23] and the references therein).…”