Minimizing Maximum Unmet Demand by Transportations Between Adjacent Nodes Characterized by Supplies and Demands

Suppose we are given a graph with nodes characterized by amounts of supplies and demands of multiple commodities. The amounts of commodities stored at nodes (supplies) are given by positive numbers while those of demands at nodes are given by negative numbers. To meet demands we send commodities from nodes to neighbors by using vehicles, one at each node, with some loading capacity moving to and from neighbors. In this paper we adopt a one-way transportation model in which we just send commodities from a node to one of its neighbors along an edge. When we choose one neighbor at each node, we have a set of trips which naturally define a graph such that each connected component has at most one cycle, which is known as a pseudoforest. We present a linear-time algorithm for deciding whether there is a set of trips that meet all demands using one-way multi-commodity transportations on a pseudoforest with node degrees bounded by a constant. Using the algorithm, we first present an efficient algorithm for finding an optimal set of one-way one-commodity trips that minimize the maximum unmet demand on a pseudoforest, and then extend the idea to a multi-commodity problem on a pseudoforest with node degrees bounded by a constant.