Representative families: A unified tradeoff-based approach

Given a matroid M=(E,I), and a family S of p-subsets of E, a subfamily ͈⊆S represents S if for any X∈S and Y⊆E\X satisfying X∪Y∈I, there is a set X∈͈ disjoint from Y, where X∪Y∈I. We show that a powerful technique for computing representative families, introduced by Fomin et al. (2014) [5], leads to a unified approach for substantially improving the running times of parameterized algorithms for some classic problems. This includes k-Partial Cover, k-Internal Out-Branching, and Long Directed Cycle, among others. Our approach exploits an interesting tradeoff between running time and the representative family size.