This thesis considers a family of combinatorial problems known under the name Knapsack Problems. As all the problems are A7)-hard we are searching for exact solution techniques having reasonable solution times for nearly all instances encountered in practice, despite having exponential time bounds for a number of highly contrived problem instances. A similar behavior is known from the Simplex algorithm, which despite its exponential worst-case behavior has reasonable solution times for all realistic problems.

We present the newly developed core concept for the Multidimensional Knapsack Problem (MKP) which is an extension of the classical concept for the one-dimensional case. The core for the multidimensional problem is defined in dependence of a chosen efficiency function of the items, since no single obvious efficiency measure is available for MKP. An empirical study on the cores of widely-used benchmark instances is presented, as well as experiments with different approximate core sizes. Furthermore we describe a memetic algorithm and a relaxation guided variable neighborhood search for the MKP, which are applied to the original and to the core problems. The experimental results show that given a fixed run-time, the di#erent metaheuristics as well as a general purpose integer linear programming solver yield better solution when applied to approximate core problems of fixed size.

We study the multidimensional knapsack problem, present some theoretical and empirical results about its structure, and evaluate different Integer Linear Programming (ILP) based, metaheuristic, and collaborative approaches for it. We start by considering the distances between optimal solutions to the LP-relaxation and the original problem and then introduce a new core concept for the MKP, which we study extensively. The empirical analysis is then used to develop new concepts for solving the MKP using ILP-based and memetic algorithms. Different collaborative combinations of the presented methods are discussed and evaluated. Further computational experiments with longer run-times are also performed in order to compare the solutions of our approaches to the best known solutions of another so far leading approach for common MKP benchmark instances. The extensive computational experiments show the effectiveness of the proposed methods, which yield highly competitive results in significantly shorter run-times than previously described approaches.

We consider an extension of the 0-1 multidimensional knapsack problem in which there are greater-than-equal inequalities, called demand constraints, besides the standard less-than-equal constraints. Moreover the objective function coefficients are not constrained in sign. This problem is worth considering because it is embedded in the models of practical applications, it has an intriguing combinatorial structure and it appears to be a challenging problem for commercial ILP solvers. Our approach is based on a nested tabu search algorithm in which neighborhoods of different structure are exploited. A first tabu search procedure is carried on in which mainly the infeasible region is explored. Once feasibility has been gained, a second tabu search procedure, which analyses only feasible solutions, is applied. The algorithm has been tested on a wide set of instances. Computational results are discussed.

Regression testing is frequently performed in a time constrained environment. This paper explains how 0/1 knapsack solvers (e.g., greedy, dynamic programming, and the core algorithm) can identify a test suite reordering that rapidly covers the test requirements and always terminates within a specified testing time limit. We conducted experiments that reveal fundamental trade-offs in the (i) time and space costs that are associated with creating a reordered test suite and (ii) quality of the resulting prioritization. We find knapsack-based prioritizers that ignore the overlap in test case coverage incur a low time overhead and a moderate to high space overhead while creating prioritizations exhibiting a minor to modest decrease in effectiveness. We also find that the most sophisticated 0/1 knapsack solvers do not always identify the most effective prioritization, suggesting that overlap-aware prioritizers with a higher time overhead are useful in certain testing contexts.