Partitioning trees of supply and demand

Assume that a tree T has a number n_s of "supply vertices" and all the other vertices are "demand vertices." Each supply vertex is assigned a positive number called a supply, while each demand vertex is assigned a positive number called a demand. One wishes to partition T into exactly n_s subtrees by deleting edges from T so that each subtree contains exactly one supply vertex whose supply is no less than the sum of demands of all demand vertices in the subtree. The "partition problem" is a decision problem to ask whether T has such a partition. The "maximum partition problem" is an optimization version of the partition problem. In this paper, we give three algorithms for the problems. The first is a linear-time algorithm for the partition problem. The second is a pseudopolynomial-time algorithm for the maximum partition problem. The third is a fully polynomial-time approximation scheme (FPTAS) for the maximum partition problem.