A Note on the Decomposition of Poly-linking Systems and the Minors of Generalized Polymatroids

A NOTE ON THE DECOMPOSITION OF POLY-LINKING SYSTEMS AND THE MINORS OF GENERALIZED POLYMATROIDS Masataka Nakamura University of Tokyo (Received June 26, 1987; Revised April 12, 1988) In this note we shall describe the results obtained by applying the decomposition principle established in [2], [3] to poly·linking systems. In order to state those results in a logically self-consistent way, we shall introduce a new notion of 'minors' of generalized polymatroids. Firstly, it is observed that the subsystems dermed from a poly-linking system through our decomposition method are not in general poly-linking systems any more. This difficulty can be overcome by considering a polylinking system as a special case of generalized polymatroids. In fact, it can be shown by an easy calculation that a poly-linking system is equivalent to a special case of generalized polymatroids. And When a poly-linking system is considered as a generalized polymatroid, its resultant subsystems through our decomposition method are seen to be the 'minors' of this generalized polymatroid. The notion of a 'minor' of a generalized polymatroid is first introduced in this paper. Hence from the point of view of our decomposition prinCiple, the notion of 'poly-linking system' is not self-consistent, and it should be treated as a special case of generalized polymatroids. Secondly, as a direct consequence of the results in [2], [3], a direct-sum decomposition of the optimal solutions of the intersection problems on poly-linking systems is induced. Lastly, we shall investigate the parametrized type of the intersection problem on poly-linking systems where the rank functions are multiplied by positive real parameters.