# Knapsack Problems

## Hans Kellerer

Language: English

Pages: 548

ISBN: 3540402861

Format: PDF / Kindle (mobi) / ePub

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.

programming by profits. The main idea of this version is to reach every possible total profit value with a subset of items of minimal total weight. Clearly, the highest total profit value, which can be reached by a subset of weight not greater than the capacity c, will be an optimal solution. This concept is called dynamic programming by reaching. It will be used in Section 2.6 to construct a special type of approximation algorithm for the knapsack problem. Since this version of dynamic

of P are the inequalities x i ~ 0 for j = 1, ... , n. This can be seen by removing ei from V, i.e. considering V \ {ei }. All remaining vectors are elements of F = {x E P I xi = O} and affinely independent. 3.10 The Knapsack Polytope 69 The study of the facets of the knapsack polytope dates back to Balas [18], Hammer, Johnson and Peled [207] and Wolsey [488] who investigated an important class of valid inequalities for the knapsack problem, the so-called minimal cover inequalities: Let S ~ N

core size is difficult to estimate in advance, Pisinger [381] proposed to use an expanding core algorithm, which simply starts with a core consisting of the split item only, and then adds more items to the core when needed. Using branch-and-bound for the enumeration, one gets the Expknap algorithm outlined in Figure 5.11. Since sorting and reduction are only done when needed, no more effort will be made than absolutely necessary. Also the reductions are tighter when postponed in this way, since

(see Section 2.3) on the scaled profit space. This means that y(q) = w signifies the existence of a subset of items with total profit equal to q ·IE/n and weight sum w. An algorithmic description of the resulting Basic-FPTAS is given in Figure 6.2. Exactly the same arguments used for showing Theorem 2.6.6 also prove that BasicFPTAS is indeed an FPTAS for (KP). Note that condition (2.19) from Section 2.6 is valid also for the scaling factor IE/n since Prnax ::; I. The running time of BasicFPTAS is

arose in many practical optimization problems, some of the more general variants of (KP) have become standard problems of their own. We will introduce several of them in the following section and deal with many others in the later chapters of this book. Beside these explicit occurrences of knapsack problems it should be noted that many solution methods of more complex problems employ the knapsack problem (sometimes iteratively) as a subproblem. Therefore, a comprehensive study of the knapsack