Before revision:  Our approach is based on solving a different fractional relaxation than that of Alon et al. , where we combine dual fitting and multiplicative weight updates approaches .
After revision:  This is an asymptotic improvement in a typical scenario where |F| \ll |C|. We achieve this by a more direct approach: we design an algorithm for a fractional relaxation of the non-metric facility location problem with clustered facilities. To handle the constraints of such non-covering LP, we combine the dual fitting and multiplicative weight updates approaches .