Approximative Algorithms for Discrete Optimization Problems

Abstract

Publisher Summary The general discrete optimization problem is defined over an arbitrary finite set S on which a mapping into the real numbers is defined. S will be called the set of feasible solutions or the feasible set. In order to provide problem with more algebraic structure, one assumes for combinatorial optimization problems that the set of feasible solutions is a subset of the power set of a given finite set. This enables us to define intersections and unions of elements of S and to identify these elements by a (0, 1)-incidence vector of length |E|. In case of combinatorial optimization problems, only separable objective functions are considered. Most well known combinatorial optimization problems and the classical integer programming problem can be stated in the general form with additional assumptions. One advantage of decision problems (or “yes-no-problems”) of type is that they are canonically related to language recognition problems on turing machines, in that the set P corresponds to the set of languages accepted by the turing machine.