Let $M=(E,{ \cal I})$ be a matroid, and let $\cal S$ be a family of subsets of size $p$ of $E$. A subfamily $\widehat{ \cal S}\subseteq{ \cal S}$ represents ${ \cal S}$ if for every pair of sets $X\in{ \cal S}$ and $Y\subseteq E\setminus X$ such that $X\cup Y\in{ \cal I}$, there is a set $\widehat{ X}\in\widehat{ \cal S}$ disjoint from $Y$ such that $\widehat{ X}\cup Y\in{ \cal I}$. 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.
Categories: cs.DS