An O(pn + 1.151p)-algorithm for p-Profit Cover and its practical implications for vertex cover

Abstract

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 G E has an integer weight w(e) > 0, and the profit is determined by Σ/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. © Springer-Verlag Berlin Heidelberg 2002.