Generalized p-Center Problems: Complexity Results and Approximation Algorithms

In an earlier paper [Hud91], two "alternative" p-Center problems, where the centers serving customers must be chosen so that exactly one node from each of p prespecified disjoint pairs of nodes is selected, were shown to be NP-complete. This paper considers a generalized version of these problems, in which the nodes from which the p servers are to be selected are partitioned into k sets and the number of servers selected from each set must be within a prespecified range. We refer to these problems as the "Set" p-Center problems. We establish that the triangle inequality (\Delta-inequality) versions of these problems, in which the edge weights are assumed to satisfy the triangle inequality, are also NP-complete. We also provide a polynomial time approximation algorithm for the two \Delta-inequality "Set" p-Center problems that is optimal for one of the problems in the sense that no algorithm with polynomial running time can provide a better constant factor performance guarantee, unless ..