Compact Location Problems with Budget and Communication Constraints

We consider the problem of placing a speciied number p of facilities on the nodes of a given network with two nonnegative edge{ weight functions so as to minimize the diameter of the placement with respect to the rst weight function subject to a diameter{ or sum{ constraint with respect to the second weight function. Deene an (;){approximation algorithm as a polynomial{time algorithm that produces a solution within times the optimal value with respect to the rst weight function, violating the constraint with respect to the second weight function by a factor of at most. We show that in general obtaining an (;){approximation for any xed ; 1 is NP{hard for any of these problems. We also present eecient approximation algorithms for several of the problems studied, when both edge{weight functions obey the triangle inequality.