Parametric power supply networks

Abstract

Suppose that each vertex of a graph $G$ is either a supply vertex or a demand vertex and is assigned a supply or a demand. All demands and supplies are nonnegative constant numbers in a steady network, while they are functions of a variable $\lambda$ in a parametric network. Each demand vertex can receive “power” from exactly one supply vertex through edges in $G$. One thus wishes to partition $G$ to connected components by deleting edges from $G$ so that each component has exactly one supply vertex whose supply is at least the sum of demands in the component. The “partition problem” asks whether $G$ has such a partition. If $G$ has no such partition, one wishes to find the maximum number $r^*$, $0\le r^* <1$, such that $G$ has such a partition when every demand is reduced to $r^*$ times the original demand. The “maximum supply rate problem” asks to compute $r^*$. In this paper, we deal with a network in which $G$ is a tree, and first give a polynomial-time algorithm for the maximum supply rate problem for a steady tree network, and then give an algorithm for the partition problem on a parametric tree network, which takes pseudo-polynomial time if all the supplies and demands are piecewise linear functions of $\lambda$.