Let M = (E,I) be a matroid, and let S be a family of subsets of size p of E. A subfamily S is contained in S represents S if for every pair of sets X ∈ S and Y is contained in E X such that X∪Y ∈ I, there is a set X ∈ S disjoint from Y such that X ∈ Y ∈ I. Fomin et al. (Proc. ACM-SIAM Symposium on Discrete Algorithms, 2014) introduced a powerful technique for fast computation of representative families for uniform matroids. In this paper, we show that this technique leads to a unified approach for substantially improving the running times of parameterized algorithms for some classic problems. This includes, among others, k-Partial Cover, k-Internal Out-Branching, and Long Directed Cycle. Our approach exploits an interesting tradeoff between running time and the size of the representative families.
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