Complexity results on planar multifacility location problems with forbidden regions

Abstract

In this paper we deal with the planar location problem with forbidden regions. We consider the median objective with block norms and show that this problem is APX-hard, even when considering the Manhattan metric as distance function and polyhedral forbidden areas. As direct consequence, the problem cannot be approximated in polynomial time within a factor of 1.0019, unless $$P=NP$$ . In addition, we give a dominating set that contains at least one optimal solution. Based on this result an approximation algorithm is derived. For special instances it is possible to improve the algorithm. These instances include problems with bounded forbidden areas and a special structure as interrelation between the new facilities. For uniform weights, this algorithm becomes an FPTAS.