Kombinatorische optimierung in halbgruppen

Abstract

By an algebraisation of the objective function is achieved that combinatorial optimization problems with different kinds of objective functions (e.g. sums or bottleneck objective functions) now occur as special cases of one general problem. The algebraisation respects not only the structure of the underlying problems but also the structure of algorithms for solving these problems. Therefore the generalized problems belong to the same complexity class as the original problems.

Combinatorial optimization problems, which can be formulated without real variables (e.g. assignment problems), can be considered now in totally ordered semigroups, where (S,*,≤) obeys additionally a strong combatibility axiom and a divisibility axiom. For problems with real variables some additional combatibility axioms between the domain ω of the variables and the semigroup S have to be fulfilled.

After the investigation of the structure of the systems (S,*,≤) and (S,*,≤;ω) general assignment problems and maximal flow problems in networks with generalized costs are considered and algorithms are given for solving these problems.