Approximating k-median with non-uniform capacities

In this paper we give a constant factor approximation algorithm for the capacitated k-median problem. Our algorithm produces a solution where capacities are exceeded by at most a constant factor, while the number of open facilities is at most k. This problem resisted attempts to apply the plethora of methods designed for the uncapacitated case. Our algorithm is based on adding some new ingredients to the approach using the primal-dual schema and lagrangian relaxations. Previous results on the capacitated k-median problem gave approximations where the number of facilities is exceeded by some constant factor. Relaxing the constraint on the number of facilities seems to render k-median problems much simpler. In some applications it is important not to violate the constraint on the number of facilities, whereas relaxing the capacity constraints is a natural thing to do, as the capacities express rough estimates on cluster sizes.