We consider the directed Steiner tree problem (DSTP) with a constraint on the total number of arcs (hops) in the tree. This problem is known to be NP-hard, and therefore, only heuristics can be applied in the case of its large-scale instances. For the hop-constrained DSTP, we propose local search strategies aimed at improving any heuristically produced initial Steiner tree. They are based on solving a sequence of hop-constrained shortest path problems for which we have recently developed efficient label correcting algorithms. The presented approach is applied to finding suitable 3D locations where unmanned aerial vehicles (UAVs) can be placed to relay information gathered in multi-target monitoring and surveillance. The efficiency of our algorithms is illustrated by results of numerical experiments involving problem instances with up to 40 000 nodes and up to 20 million arcs.
Technical Report LiTH-MAT-R–2014/10–SE, Department of Mathematics, Linkoping University, Sweden, August/2014