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.