An O(pn + 1.151p)-Algorithm for p-Profit Cover and Its Practical Implications for Vertex Cover

We introduce the problem Profit Cover which finds application in, among other areas, psychology of decision-making. A common assumption is that net value is a major determinant of human choice. Profit Cover incorporates the notion of net value in its definition. For a given graph G = (V,E) and an integer p > 0, the goal is to determine PC ⊆ V such that the profit, |E′| − |PC|, is at least p, where E′ are the by PC covered edges. We show that p-Profit Cover is a parameterization of Vertex Cover. We present a fixed-parameter-tractable (fpt) algorithm for p-Profit Cover that runs in O(p|V |+1.150964p). The algorithm generalizes to an fpt-algorithm of the same time complexity solving the problem p-Edge Weighted Profit Cover, where each edge e ∈ E has an integer weight w(e) > 0, and the profit is determined by ∑ e∈E′ w(e)−|PC|. We combine our algorithm for p-Profit Cover with an fpt-algorithm for k-Vertex Cover. We show that this results in a more efficient implementation to solve Minimum Vertex Cover than each of the algorithms independently.