Parameterized Algorithms for Red-Blue Weighted Vertex Cover on Trees

\textproc{Weighted Vertex Cover} is a variation of an extensively studied NP-complete problem, \textproc{Vertex Cover}, in which we are given a graph, $G = (V,E,w)$, where function $w:V \rightarrow \mathbb{Q}^{+}$ and a parameter $k$. The objective is to determine if there exists a vertex cover, $S$, such that $\sum_{v \in S}w(v) \leq k$. In our work, we first study the hardness of \textproc{Weighted Vertex Cover} and then examine this problem under parameterization by $l$ and $k$, where $l$ is the number of vertices with fractional weights. Then, we study the \textproc{Red-Blue Weighted Vertex Cover} problem on trees in detail. In this problem, we are given a tree, $T=(V,E,w)$, where function $w:V \rightarrow \mathbb{Q}^{+}$, a function $c:V \rightarrow \{R,B\}$ and two parameters $k$ and $k_R$. We have to determine if there exists a vertex cover, $S$, such that $\sum_{v \in S}w(v) \leq k$ and $\sum_{\substack{v \in S\\ c(v) = R}}w(v) \leq k_R$. We tackle this problem by applying different reduction techniques and meaningful parameterizations. We also study some restrictive versions of this problem.