Collective Fast Delivery by Energy-Efficient Agents

We consider k mobile agents initially located at distinct nodes of an undirected graph (on n nodes, with edge lengths). The agents have to deliver a single item from a given source node s to a given target node t. The agents can move along the edges of the graph, starting at time 0, with respect to the following: Each agent i has a weight omega_i that defines the rate of energy consumption while travelling a distance in the graph, and a velocity upsilon_i with which it can move. We are interested in schedules (operating the k agents) that result in a small delivery time T (time when the item arrives at t), and small total energy consumption E. Concretely, we ask for a schedule that: either (i) Minimizes T, (ii) Minimizes lexicographically (T,E) (prioritizing fast delivery), or (iii) Minimizes epsilon * T + (1-epsilon)* E, for a given epsilon in (0,1). We show that (i) is solvable in polynomial time, and show that (ii) is polynomial-time solvable for uniform velocities and solvable in time O(n+k log k) for arbitrary velocities on paths, but in general is NP-hard even on planar graphs. As a corollary of our hardness result, (iii) is NP-hard, too. We show that there is a 2-approximation algorithm for (iii) using a single agent.