Representative Families: A Unified Tradeoff-Based Approach

TL;DR

This paper introduces a unified tradeoff-based approach for efficiently computing representative families in matroids, significantly improving parameterized algorithms for problems like k-Partial Cover and Long Directed Cycle.

Contribution

It generalizes a technique for uniform matroids to a broader framework, enhancing algorithmic efficiency through a tradeoff between runtime and family size.

Findings

Improved algorithms for k-Partial Cover and Long Directed Cycle

Unified framework for representative families in matroids

Tradeoff between running time and family size

Abstract

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.