Showing papers by "David Pisinger published in 2004"

Introduction to NP-Completeness of Knapsack Problems

Abstract: The reader may have noticed that for all the considered variants of the knapsack problem, no polynomial time algorithm have been presented which solves the problem to optimality. Indeed all the algorithms described are based on some kind of search and prune methods, which in the worst case may take exponential time. It would be a satisfying result if we somehow could prove it is not possible to find an algorithm which runs in polynomial time, somehow having evidence that the presented methods are “as good as we can do”. However, no proof has been found showing that the considered variants of the knapsack problem cannot be solved to optimality in polynomial time.

Multidimensional Knapsack Problems

Abstract: In this first chapter of extensions and generalizations of the basic knapsack problem (KP) we will add additional constraints to the single weight constraint (1.2) thus attaining the multidimensional knapsack problem. After the introduction we will deal extensively with relaxations and reductions in Section 9.2. Exact algorithms to compute optimal solutions will be covered in Section 9.3 followed by results on approximation in Section 9.4. A detailed treatment of heuristic methods will be given in Section 9.5. Separate sections are devoted to two special cases, namely the two-dimensional knapsack problem (Section 9.6) and the cardinality constrained knapsack problem (Section 9.7). Finally, we will consider the combination of multiple constraints and multiple-choice selection of items from classes (see Chapter 11 for the one-dimensional case) in Section 9.8.

The Multiple-Choice Knapsack Problem

Abstract: The multiple-choice knapsack problem (MCKP) is a generalization of the ordinary knapsack problem, where the set of items is partitioned into classes. The binary choice of taking an item is replaced by the selection of exactly one item out of each class of items. In Section 7.1 we already noticed that a (BKP) can be formulated as a (MCKP), and indeed the (MCKP) model is one of the most flexible knapsack models. (MCKP) is also denoted as knapsack problem with generalized upper bound constraints or for short knapsack problem with GUB.

Scheduling Transportation of Live Animals to Avoid the Spread of Diseases

Abstract: In the classical vehicle-routing problem (VRP) the objective is to service some geographically scattered customers with a given number of vehicles at the minimal cost. In the present paper, we consider a variant of the VRP where the vehicles should deliver some goods between groups of customers. The customers have an associated time window, a precedence number, and a quantity. Each vehicle should visit the customers within their time windows, in nondecreasing order of precedence respecting the capacity of the vehicle. The problem will be denoted the pickup-and-delivery problem with time windows and precedence constraints (PDPTWP). The PDPTWP has applications in the transportation of live animals where veterinary rules demand that the livestocks are visited in a given sequence in order not to spread specific diseases. We propose a tighter formulation of the PDPTWP based on Dantzig-Wolfe decomposition. The formulation splits the problem into a master problem, which is a kind of set-covering problem, and a subproblem that generates legal routes for a single vehicle. The LP-relaxation of the decomposed problem is solved through delayed column generation. Computational experiments show that the obtained bounds are less than 0.24% from optimum for the considered problems. As solving the pricing problems takes up the majority of the solution time, a reformulation of the problem is proposed that makes use of the precedence constraints. By merging customers having the same precedence number into "super nodes," the pricing problem may be reformulated as a shortest-path problem defined on an acyclic layered graph. This makes it possible to solve the pricing problem in pseudopolynomial time through dynamic programming. The paper concludes with a comprehensive computational study involving real-life instances from the transportation of live pigs. It is demonstrated that instances with up to 580 nodes can be solved to optimality.

Multiple Knapsack Problems

Abstract: The multiple knapsack problem is a generalization of the standard knapsack problem (KP) from a single knapsack to m knapsacks with (possibly) different capacities. The objective is to assign each item to at most one of the knapsacks such that none of the capacity constraints are violated and the total profit of the items put into knapsacks is maximized.

The Subset Sum Problem

Abstract: Given a set N = {1,..., n} of n items with positive integer weights w1,..., w n and a capacity c, the subset sum problem (SSP) is to find a subset of N such that the corresponding total weight is maximized without exceeding the capacity c. Recall the formal definition as introduced in Section 1.2: $$ (SSP)\,\max {\rm{imize}}\;\sum\limits_{j = 1}^n {{w_j}{x_j}} $$ (4.1) $$ {\rm{subject}}\;{\rm{to}}\;\sum\limits_{j = 1}^n {{w_j}{x_j}} \le c,$$ (4.2) $$ {x_j} \in \left\{ {0,1} \right\},\;j = 1,...,n.$$ (4.3)

The Quadratic Knapsack Problem

Abstract: In all the variants of the knapsack problems considered so far the profit of choosing a given item was independent of the other items chosen. In many real life applications as well as in problems with roots in graph theory it is natural to assume that the profit of a packing also should reflect how well the given items fit together. One possible formulation of such an interdependence is the quadratic knapsack problem (QKP) in which an item has a corresponding profit and an additional profit is redeemed if the item is selected together with another item. (QKP) was first introduced by Gallo, Hammer and Simeone [160] and has been studied intensively in the last decade due to its simple structure and challenging difficulty.

The Bounded Knapsack Problem

Abstract: Right from the beginning of research on the knapsack problem in the early six-ties separate considerations were devoted to problems where a number of identical copies of every item are given or even an unlimited amount of each item is available. The corresponding problems are known as the bounded and unbounded knapsack problem, respectively. Since there exists a considerable amount of theoretical, algorithmic and computational results which apply for only one of these two problems, we found it appropriate to deal with them in separate chapters. Hence, we restrict this chapter to the bounded case whereas the unbounded case will be treated in Chapter 8.

Other Knapsack Problems

Abstract: In this chapter we consider knapsack type problems which have not been investigated in the preceding chapters. There is a huge amount of different kinds of variations of the knapsack problem in the scientific literature, often a specific problem is treated in only one or two papers. Thus, we could not include every knapsack variant but we tried to make a representative selection of interesting problems. Two problems will be presented in the first two sections more detailed. In Section 13.1 we start with multiobjective knapsack problems and and continue in Section 13.2 with results about precedence constraint knapsack problems. Finally, Section 13.3 contains a collection of several other variations of knapsack problems and the main results for these problems are mentioned.

The Unbounded Knapsack Problem

Abstract: The availability of an unlimited number of copies for every item type leads to phenomena which are quite different from the bounded case described in Chapter 7. Although there is a natural bound of how many copies of any item type can fit into a knapsack the structure of the problem is in several aspects not the same as for the case with a prespecified bound. Hence, it is worthwhile to devote this separate chapter to the unbounded knapsack problem (UKP).

Approximation Algorithms for the Knapsack Problem

Abstract: Approximation algorithms and in particular approximation schemes like PTAS and FPTAS were already introduced in Section 2.5 and 2.6, respectively. The main motivation in these sections was to illustrate the basic concept of constructing simple approximation schemes. The focus was put on algorithms where both the correctness and the required complexities were easy to understand without having to go deeply into the details of complicated technical constructions. Hence, an intuitive understanding about the basic features of approximation should have been brought to the reader which is a necessary prerequisite to tackle the more sophisticated methods required to improve upon the performance of these simple algorithms.

Exact Solution of the Knapsack Problem

Abstract: Assume that a set of n items is given, each item j having an integer profit p j and an integer weight w j . The knapsack problem asks to choose a subset of the items such that their overall profit is maximized, while the overall weight does not exceed a given capacity c. Introducing binary variables x j to indicate whether item j is included in the knapsack or not the model may be defined: $$ {\rm{(KP)}}\,{\rm{maximize}}\;\sum\limits_{j = 1}^n {{p_j}{x_j}} $$ (5.1) $$ {\rm{subject}}\;{\rm{to}}\;\sum\limits_{j = 1}^n {{w_j}{x_j}} \le c,$$ (5.2) $$ {x_j} \in \left\{ {0,1} \right\},\;j = 1,...,n.$$ (5.3)

Basic Algorithmic Concepts

Abstract: In this chapter we introduce some algorithmic ideas which will be used in more elaborate ways in various chapters for different problems. The aim of presenting these ideas in a simple form is twofold. On one hand we want to provide the reader, who is a novice in the area of knapsack problems or combinatorial and integer programming in general, with a basic introduction such that no other reference is needed to work successfully with the more advanced algorithms in the remainder of this book. On the other hand, we would like to establish a common conceptual cornerstone from which the other chapters can develop the more refined variants of these basic ideas as they are required for each of the individual problems covered there. In this way we can avoid repetitions without affecting the self-contained character of the remaining chapters.