Exact and approximate algorithms for movement problems on (special classes of) graphs

When a large collection of objects (e.g., robots, sensors, etc.) has to be deployed in a given environment, it is often required to plan a coordinated motion of the objects from their initial position to a final configuration enjoying some global property. In such a scenario, the problem of minimizing some function of the distance travelled, and therefore of reducing energy consumption, is of vital importance. In this paper we study several motion planning problems that arise when the objects initially sit on the vertices of a graph, and they must be moved so as that the final vertices that receive (at least) one object induce a subgraph enjoying a given property. In particular, we consider the notable properties of connectivity, independence, completeness, and finally that of being a vertex-cutset w.r.t. a pair of fixed vertices. We study these problems with the aim of minimizing a number of natural measures, namely the average/overall distance travelled, the maximum distance travelled, and the number of objects that need to be moved. To this respect, we provide several approximability and inapproximability results, most of which are tight.