Approximation algorithm for the k- product uncapacitated facility location problem

We consider one kind of uncapacitated facility location problem which we call k-product uncapacitated facility location problem with no-fixed costs(k-PUFLPN). The problem can be defined as follows: There is a set of demand points where clients are located and a set of potential sites where facilities of unlimited capacities can be set up. There are k different kinds of products. Each client needs to be supplied with k kinds of products by a set of k different facilities and each facility can be set up to supply only a distinct product with no fixed cost. There is a non-negative cost of shipping goods between each pair of locations. These costs are assumed to be symmetric and satisfy the triangle inequality. We want to select a set of facilities to be opened and their designated products and to find an assignment for each client to a set of k facilities so as to minimize the sum of the shipping costs. In this paper, we propose an approximation algorithm with a performance guarantee of (3/2) k -1 for the k-PUFLPN.