of course identical, respectively, to those of user u and we have ˜ f $^{wsr}$( G ) = f $^{wsr}$( G ) , ∀ G ∈ J . Notice that under the admission control assumtion each member of I is feasible so that J = I . Further, each member in ˜ I is also feasible. Then, we can verify from the arguments used to prove Theorem 2 that ˜ f $^{wsr}$( . ) is a normalized non-negative submodular set function over ˜ I . With this understanding, we can re-formulate (1) as the following constrained set function maximization, Then, let ˆ G be the set returned by Algorithm II and let ˆ O denote any optimal solution of (27). Notice that ˆ G ∪ ˆ O ∈ ˜ I so that the extended set function is defined and is submodular over all subsets of ˆ G ∪ ˆ O . This enables us to invoke the arguments presented in [23] to prove the aproximation guarantee and thereby establish our desired result. Corollary 4. In the absence of per-user minimum rate constraints the problem in (1) reduces to the monotone submodular maximization problem with a matroid constraint, for which the greedy stage output itself (and thus Algorithm II) yields an approximation factor of 1 / 2 . Regarding the worst-case complexity of Algorithm II, we can show that it scales polynomially in | Ω | /ϵ [23] . In practice, in all of our simulation runs we observed that with the greedy stage initialization the enhanced LS stage converges very quickly. Moreover, the implementation of the greedy stage of Algorithm II can be significantly improved by exploiting the submodularity of f $^{wsr}$( . ) over J , as done in the lazy greedy implementation [24]. Let us now focus on the problem in (2). In order to design an approximation algorithm, we consider the problem in (28), where we recall our convention that 0 ln(0) = 0 . Note that (28) imposes an orthogonal split on (2) and allows for each user to be associated to (and served by) exactly one node. The problem in (28) has been widely considered before and seeks to optimize the PF utility over user associations but does not permit dual connectivity. There are several approaches to solve (28), including an efficient optimal one [5] and approximately optimal ones with lower complexity [3], [5], [22]. In Algorithm III we propose a method to solve (2) where we can leverage any of the available approaches to solve (28). Once a user association is so obtained, we enhance it by exploiting dual connectivity. Hence, all users associated to any pico max G∈ I { ˜ f $^{wsr}$( G ) } (27)