Improved approximation algorithms for uncapacitated facility location

We consider the uncapacitated facility location problem. In this problem, there is a set of locations at which facilities can be built; a fixed cost f(i) is incurred if a facility is opened at location i. Furthermore, there is a set of demand locations to be serviced by the opened facilities; if the demand location j is assigned to a facility at location i, then there is an associated service cost of c(ij). The objective is to determine which facilities to open and an assignment of demand points to the opened facilities, so as to minimize the total cost. We assume that the service costs c(ij) are symmetric and satisfy the triangle inequality. For this problem we obtain a (1 + 2/e)-approximation algorithm, where 1 + 2/e approximate to 1.736, which is a significant improvement on the previously known approximation guarantees. The algorithm works by rounding an optimal fractional solution to a linear programming relaxation. Our techniques use properties of optimal solutions to the linear program, randomized rounding, as well as a generalization of the decomposition techniques of Shmoys, Tardos, and Aardal.