Parameterized Algorithms for Red-Blue Weighted Vertex Cover on Trees
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Weighted Vertex Cover is a variation of an extensively studied NP-complete problem, Vertex Cover, in which we are given a graph, G = (V,E,w), where function w:V →Q^+ and a parameter k. The objective is to determine if there exists a vertex cover, S, such that ∑_v ∈ Sw(v) ≤ k. In our work, we first study the hardness of 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 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 →Q^+, a function c:V →{R,B} and two parameters k and k_R. We have to determine if there exists a vertex cover, S, such that ∑_v ∈ Sw(v) ≤ k and ∑_v ∈ S c(v) = Rw(v) ≤ k_R. We tackle this problem by applying different reduction techniques and meaningful parameterizations. We also study some restrictive versions of this problem.
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