Objective functions with redundant domains

Abstract

Let (E, A) be a set system consisting of a finite collection A of subsets of a ground set E, and suppose that we have a function φ which maps A into some set S . Now removing a subset K from E gives a restriction A( K) to those sets of A disjoint from K, and we have a corresponding restriction φ| A( K) of our function φ. If the removal of K does not affect the image set of φ, that is Im(φ| A( X) ) = Im(φ), then we will say that K is a kernel set of A with respect to φ. Such sets are potentially useful in optimisation problems defined in terms of φ. We will call the set of all subsets of E that are kernel sets with respect to φ a kernel system and denote it by Ker φ (A). Motivated by the optimisation theme, we ask which kernel systems are matroids. For instance, if A is the collection of forests in a graph G with coloured edges and φ counts how many edges of each colour occurs in a forest then Ker φ (A) is isomorphic to the disjoint sum of the cocycle matroids of the differently coloured subgraphs; on the other hand, if A is the power set of a set of positive integers, and φ is the function which takes the values 1 and 0 on subsets according to whether they are sum-free or not, then we show that Ker φ (A) is essentially never a matroid.