We study Connected Facility Location problems. We are given a connected graph G = (V, E) with nonnegative edge cost ce for each edge e E, a set of clients D V such that each client j D has positive demand dj and a set of facilities F V each has nonnegative opening cost fi and capacity to serve all client demands. The objective is to open a subset of facilities, say ^F, to assign each client j D to exactly one open facility i(j) and to connect all open facilities by a Steiner tree T such that the cost i ^F fi + jD djci(j)j +M eT ce is minimized for a given