Compact location problems

We consider the problem of placing a specified number ( p) of facilities on the nodes of a network so as to minimize some measure of the distances between facilities. This type of problem models a number of problems arising in facility location, statistical clustering, pattern recognition, and processor allocation problems in multiprocessor systems. We consider the problem under three different objectives, namely minimizing the diameter, minimizing the average distance, and minimizing the variance. We observe that, in general, the problem is NP-hard under any of the objectives. Further, even obtaining a constant factor approximation for any of the objectives is NP-hard. We present a general framework for obtaining near-optimal solutions to the compact location problems for the above measures, when the distances satisfy the triangle inequality. We show that this framework can be extended to the case when there are also node weights. Further, we investigate the complexity and approximability of more general versions of these problems, where two distance values are specified for each pair of potential sites. In these cases, the goal is to a select a specified number of facilities to minimize a function of one distance metric subject to a budget constraint on the other distance metric. We present algorithms that provide solutions which are within a small constant factor of the objective value while violating the budget constraint by only a small constant factor.