A note on the decomposition of poly-linking systems and the minors of generalized polymatroids

In this note we shall describe the results obtained by applying an established decomposition principle 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 defined 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 previous results, 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.