Matroids in Parameterized Complexity and Exact Algorithms

Years and Authors of Summarized Original Work

• 2009; Marx

• 2014; Fomin, Lokshtanov, Saurabh

• 2014; Fomin, Lokshtanov, Panolan, Saurabh

• 2014; Shachnai, Zehavi

Problem Definition

In recent years matroids have been used in the fields of parameterized complexity and exact algorithms. Many of these works mainly use a computation of representative families. Let $M = (E,\mathcal{I})$ be a matroid and $\mathcal{S} =\{ S_{1},\ldots S_{t}\} \subseteq \mathcal{I}$ be a family of independent sets of size p. A subfamily $\hat{\mathcal{S}}\subseteq \mathcal{S}$ is called a q-representative family for $\mathcal{S}$ (denoted by $\hat{\mathcal{S}}\subseteq _{rep}^{q}\mathcal{S}$), if for every $Y \subseteq E$ of size at most q, if there exists a set $S \in \mathcal{S}$ disjoint from Y with $S \cup Y \in \mathcal{I}$, then there exists a set $\hat{S} \in \hat{\mathcal{S}}$ disjoint from Y with $\hat{S} \cup Y \in \mathcal{I}$. The basic algorithmic question regarding representative...