We consider the problem of delivery of a packet from a source node to a destination node in a graph, by a team of mobile robots that have very low battery capacity (two units of energy, with one unit spent each time an agent traverses an edge). The robots can collaborate to deliver the packet by exchanging energy or the packet itself whenever two of them are co-located. We study two variants of the problem: In the fixed placement variant, the robots are initially placed at some given nodes of the graph and the question is whether delivery is feasible. We prove that this problem is NP-complete. In the free placement variant, the initial positions of the robots are not fixed but a subset of nodes H of the graph is given as input together with an integer k, and the question is whether there exists a placement of k robots at nodes in H such that delivery is possible. We prove that this problem can be solved in polynomial time, by actually computing the minimum initial energy and the placement of robots at nodes in H that allows them to carry out the delivery.

(joint work with Shantanu Das, Dariusz Dereniowski, and Christina Karousatou)