The Maximum-Mean Subtree

In this paper, we define the Maximum-Mean Subtree problem on trees, an equivalent reformulation of the Fractional Prize-Collecting Steiner Tree Problem on Trees. We describe an algorithm that solves the Maximum-Mean Subtree problem, and prove that our algorithm runs in O(n) time in the worst case, improving a previous O(n log n) algorithm. 1 The Maximum-Mean Subtree Problem Given a rooted tree of nodes such that each node has a real valued profit, we are to produce a pruning of the tree that maximizes the average profit of the remaining nodes. Note that pruning a node also prunes all of its descendants. A generalization of this problem gives each node a positive real valued cost, with the original problem assigning each node a cost of 1. The overall average of a tree is the sum of the profits divided by the sum of the costs, including only unpruned nodes. In this paper we present an algorithm that solves the generalization with both profits and costs per node in time O(n). We strictly restrict costs to be positive due to the fact that negative costs result in some instances of the problem being computationally equivalent to Subset-Sum on real numbers. While profits may be negative without affecting the correctness of the result produced by our algorithm (as long as the root has positive profit), it is not clear as to what a negative profit would represent conceptually.