Theory of Principal Partitions Revisited

The theory of principal partitions of discrete systems such as graphs, matrices, matroids, and submodular systems has been developed since 1968. In the early stage of the developments during 1968–75 the principal partition was considered as a decomposition of a discrete system into its components together with a partially ordered structure of the set of the components. It then turned out that such a decomposition with a partial order on it arises from the submodularity structure pertinent to the system and it has been realized that the principal partitions are closely related to resource allocation problems with submodular structures, which are kind of dual problems. The aim of this paper is to give an overview of the developments in the theory of principal partitions and some recent extensions with special emphasis on its relation to associated resource allocation problems in order to make it better known to researchers in combinatorial optimization.