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, $G = (V,E,w)$,wherefunction$w:V \rightarrow $\mathbb{Q}$^{+}$andaparameter$k$\mathrm{.}Theobjectiveistodetermineifthereexistsavertexcover,$S$, such that ${\sum }_{v\in S}w(v)\le k$. 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 $w:V\to {\mathbb{Q}}^{+}$, a function $c:V \rightarrow {R,B}$andtwoparameters$k$and$k_R$. We have to determine if there exists a vertex cover, $S$, such that ${\sum }_{v\in S}w(v)\le k$ and ${\sum }_{\begin{array}{c}v\in S\\ c(v)=R\end{array}}w(v)\le k_R$. We tackle this problem by applying different reduction techniques and meaningful parameterizations. We also study some restrictive versions of this problem.