Objective functions with redundant domains

نویسندگان

  • Fatima Affif Chaouche
  • Carrie G. Rutherford
  • Robin Whitty
چکیده

Let (E,A) be a set system consisting of a finite collectionA 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(K̄) to those sets of A disjoint from K, and we have a corresponding restriction φ|A(K̄) of our function φ. If the removal of K does not affect the image set of φ, that is Im(φ|A(X̄)) = 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 Kerφ(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 Kerφ(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 Kerφ(A) is essentially never a matroid.

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عنوان ژورنال:
  • J. Comb. Optim.

دوره 26  شماره 

صفحات  -

تاریخ انتشار 2013