an efficient algorithm which guarantees a constant-factor approximation. We note that to show submodularity we do not follow the direct approach of proving the original definition, but instead we consider proving another cleverly obtained sufficient condition. The latter approach then requires us to characterize the (second order) change in the optimal solution of a linear program with respect to its parameters. We expect that our result and systematic derivation will have much wider applications. We note that a conference version of this report appeared in [4]. Compared to [4], here we enhance the technical results, provide complete proofs as well as expand on simulation results, discussions and new applications. Let U denote the set of users with cardinality |U| = K and let M denote the set of Macro TPs. For each Macro TP m ∈ M , let B$_{m}$ denote the set of pico TPs assigned to macro TP m . Here the set B$_{m}$ of pico TPs facilitate the macro TP m to serve its associated users. We suppose that all indices in the set of all TPs, S = M∪P , where P = ∪$_{m}$$_{∈M}$B$_{m}$ , are distinct. An illustrative schematic for DC is shown in Fig. 1. Notice that each user can be associated with any one macro and any one pico TP from the set of pico TPs assigned to the chosen macro. For each user u ∈ U and each TP b ∈ S , we define R$_{u,b}$ to be the averaged single-user or peak rate user u can get (in bits per unit resource) when it alone is served by TP b . This single-user or peak rate is a function of the slow fading parameters (e.g. path loss and shadowing) seen by user u but not the instantaneous fast-fading ones which have been averaged out. Further, this average rate can be computed under the (widely used) assumption that all the other interfering TPs are always transmitting, or under the framework of [21] that each interfering TP transmits for a given fraction of the frame duration. We note that in the in-band DC case the picos and macros share the same spectrum band whereas in the out-of-band DC case they are assigned different bands. Our results are applicable to both scenarios and only the user peak rates have to be accordingly computed. We make the mild assumption that R$_{u,b}$ > 0 , ∀ u ∈ U ,b ∈ S . We also assume that R$_{u,m}$ R$_{u,b}$ = R$_{u}$$_{′}$$_{,m}$ $_{R$_{u}$$_{′}$$_{,b}$}$, ∀ b ∈ B$_{m}$ ,m ∈ M & u = u $^{′}$. Notice that these two assumptions hold true almost surely for all typical slow fading distributions. We first formulate a network utility II. PROBLEM FORMULATION