An O(pn + 1.151^p)-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, É′ — 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+1.150964^p). 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 $\sum\limits_{e \in E'} {w\left( e \right)} - \left| {PC} \right|$. We combine our algorithm for p-Profit e∈E′ 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.

This research is supported by a UVic research grant and by NSERC grant 54194.

All graphs considered in this article are simple and undirected.