Complexity and Approximability of Certain Bicriteria Location Problems

We investigate the complexity and approximability of some location problems when two distance values are speciied for each pair of potential sites. These problems involve the selection of a speciied number of facilities (i.e. a placement of a speciied size) to minimize a function of one distance metric subject to a budget constraint on the other distance metric. Such problems arise in several application areas including statistical clustering, pattern recognition and load{balancing in distributed systems. We show that, in general, obtaining placements that are near-optimal with respect to the rst distance metric is NP{ hard even when we allow the budget constraint on the second distance metric to be violated by a constant factor. However, when both the distance metrics satisfy the triangle inequality, we present approximation algorithms that produce placements which are near-optimal with respect to the rst distance metric while violating the budget constraint only by a small constant factor. We also present polynomial algorithms for these problems when the underlying graph is a tree.