Efficient computation of representative sets with applications in parameterized and exact algorithms

Let M = (E, I) be a matroid and let S = {S1,..., St} be a family of subsets of E of size p. A subfamily S ̂ ⊆ S is q-representative for S if for every set Y ⊆ E of size at most q, if there is a set X ∈ S disjoint from Y with X ∪ Y ∈ I, then there is a set X ̂ ∈ S ̂ disjoint from Y with X ̂ ∪ Y ∈ I. By the classical result of Bollobás, in a uniform matroid, every family of sets of size p has a q-representative family with at most p+