Collaborative Delivery with Energy-Constrained Mobile Robots

We consider the problem of collectively delivering some message from a specified source to a designated target location in a graph, using multiple mobile agents. Each agent has a limited energy which constrains the distance it can move. Hence multiple agents need to collaborate to move the message, each agent handing over the message to the next agent to carry it forward. Given the positions of the agents in the graph and their respective budgets, the problem of finding a feasible movement schedule for the agents can be challenging. We consider two variants of the problem: in non-returning delivery, the agents can stop anywhere; whereas in returning delivery, each agent needs to return to its starting location, a variant which has not been studied before. We first provide a polynomial-time algorithm for returning delivery on trees, which is in contrast to the known (weak) NP-hardness of the non-returning version. In addition, we give resource-augmented algorithms for returning delivery in general graphs. Finally, we give tight lower bounds on the required resource augmentation for both variants of the problem. In this sense, our results close the gap left by previous research.Comment: 19 pages. An extended abstract of this paper was published at the 23rd International Colloquium on Structural Information and Communication Complexity 2016, SIROCCO'1

Polygon-Constrained Motion Planning Problems

We consider the following class of polygon-constrained motion planning problems: Given a set of k centrally controlled mobile agents (say pebbles) initially sitting on the vertices of an n -vertex simple polygon P , we study how to plan their vertex-to-vertex motion in order to reach with a minimum (either maximum or total) movement (either in terms of number of hops or Euclidean distance) a final placement enjoying a given requirement. In particular, we focus on final configurations aiming at establishing some sort of visual connectivity among the pebbles, which in turn allows for wireless and optical intercommunication. Therefore, after analyzing the notable (and computationally tractable) case of gathering the pebbles at a single vertex (i.e., the so-called rendez-vous), we face the problems induced by the requirement that pebbles have eventually to be placed at: (i) a set of vertices that form a connected subgraph of the visibility graph induced by P , say G(P) (connectivity), and (ii) a set of vertices that form a clique of G(P) (clique-connectivity). We will show that these two problems are actually hard to approximate, even for the seemingly simpler case in which the hop distance is considered