On the Complexity of Some Common Geometric Location Problems

Given $n$ demand points in the plane, the $p$-center problem is to find $p$ supply points (anywhere in the plane) so as to minimize the maximum distance from a demand point to its respective nearest supply point. The $p$-median problem is to minimize the sum of distances from demand points to their respective nearest supply points. We prove that the $p$-center and the $p$-median problems relative to both the Euclidean and the rectilinear metrics are NP-hard. In fact, we prove that it is NP-hard even to approximate the $p$-center problems sufficiently closely. The reductions are from 3-satisfiability.