A Fast Algorithm for Powerful Alliances in Trees

Given a graph G = (V,E) with a positive weight function w on the vertices of G, a global powerful alliance of G is a subset S of V such that for every vertex v at least half of the total weight in the closed neighborhood of v is contributed by the vertices of S. Finding the smallest such set in general graphs is NP-complete, even when the weights are all the same. In this paper, we give a linear time algorithm that finds the smallest global powerful alliance of any weighted tree T = (V,E).