Partitioning graphs of supply and demand

Abstract

Assume 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 through edges in $G$. 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 $G$ does not have such a 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 strongly NP-hard for general graphs. In this paper, we show that the problem can be solved in pseudo-polynomial time for series–parallel graphs and partial $k$-trees–that is, graphs with bounded tree-width.