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. Further- more, 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 cij. The objective is to de- termine 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 ≈ 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 opti- mal solutions to the linear program, randomized rounding, as well as a generalization of the decomposition techniques of Shmoys, Tardos, and Aardal.