Partitioning Graphs of Supply and Demand

that each vertex of a graph G is either a supply vertex or a demand vertex and is assigned a positive integer, called a supply or a demand. Each demand vertex can receive -power- from at most one supply vertex. One thus wishes to partition G into connected components by deleting edges from G so that each component C has exactly one supply vertex whose supply is no less than the sum of demands of all demand vertices in C. If G has no such partition, one wishes to partition G into connected components so that each component C either has no supply vertex or has exactly one supply vertex whose supply is no less than the sum of demands in C, and wishes to maximize the sum of demands in all components with supply vertices. We deal with such a maximization problem, which is NP-hard even for trees and strong NP-hard for general graphs. In this paper, we give a pseudo-polynomial-time algorithm to solve the problem for series-parallel graphs. The algorithm can be easily extended for partial k-trees, that is, graphs with bounded tree-width.