Before revision:  In the robust problem,\vec{c} a parameter m is given and we need to assign some facility to at least m clients. We give polynomial-time\vec{c} constant-factor  approximation algorithms for both problems, which use standard sparsification as well as iterative rounding techniques for LP relaxations .
After revision:  In the robust problem,\vec{c} a parameter m is given and we need to connect the nearest open facility to m points in X. In this case, the service cost vector\vec{c} constant-factor approximation algorithms for both problems, which use standard sparsification as well as iterative rounding techniques for LP relaxations .