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.