Combinatorial Problems with Multiple Objectives

Abstract

The essential feature of combinatorial optimization problems is that they have a finite set of feasible solutions. This assumption has significant impact on the way we deal with these problems, both in theory and solution techniques. We shall first introduce combinatorial optimization problems and the MCO classes we consider in the chapter. Some basic observations show that in a multicriteria context combinatorial optimization is quite different from the general or linear optimization framework we have considered in earlier chapters of this text. In particular, we give a brief introduction to the concepts of computational complexity, such as NP-completeness and #P-completeness. In the subsequent sections we prove many results on computational complexity. These sections feature one selected combinatorial problem each. In addition, for each problem we present one or more solution strategies, so that the reader acquires an overview of available techniques for multicriteria combinatorial optimization. For a broader survey of the field we refer to a recent bibliography [EG00].