Variations of the prize-collecting Steiner tree problem

The prize-collecting Steiner tree problem is well known to be NP-hard. We consider seven variations of this problem generalizing several well-studied bottleneck and minsum problems with feasible solutions as trees of a graph. Four of these problems are shown to be solvable in O(m+n log n) time and the remaining are shown to be NP-hard where n is the number of nodes and m is the number of edges in the underlying graph. For one of these polynomially solvable cases, we also provide an O(m) algorithm generalizing and unifying known linear time algorithms for the bottleneck spanning tree problem, bottleneck s-t path problem, and bottleneck Steiner tree problem.