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 prespeciied 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 prespeciied range. We refer to these problems as the \Set" p-Center problems. We establish that the triangle inequality ((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-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 P=NP. For the special case \alternative" p-Center problems, which we refer to as the \Pair" p-Center problems, we extend the results in Hud91] in several ways. For example, the results mentioned above for the \Set" p-Center problems also apply to the \Pair" p-Center problems. Furthermore, we establish and exploit a correspondence between satissability and the dominating set type of problems that naturally arise when considering the decision versions of the \Pair" p-Center problems.