On the fast delivery problem with one or two packages

We study two problems where k autonomous mobile agents are initially located on distinct nodes of a weighted graph with n nodes and m edges. Each agent has a predefined velocity and can only move along the edges of the graph. The first problem is to deliver one package from a source node to a destination node. The second is to simultaneously deliver two packages, each from its source node to its destination node. These deliveries are achieved by the collective effort of the agents, which can carry and exchange a package among them. For one package, we propose an O(knlog⁡n+km) time algorithm for computing a delivery schedule that minimizes the delivery time. For two packages, we show that the problem of minimizing the maximum or the sum of the delivery times is NP-hard for arbitrary agent velocities, but polynomial-time solvable for agents with equal velocity.