f wsr 1 ( F ) ≥ ˆ O (Γ + δ, Γ + δ$_{b}$$_{1}$ e$_{b}$$_{1}$ ) + w$_{u}$$_{2}$R$_{u}$$_{2}$$_{,b}$$_{2}$ ˜ δ$_{b}$$_{2}$ + w$_{u}$$_{2}$R$_{u}$$_{2}$$_{,}$$_{1}$ ˜ δ ; f wsr 1 ( E ) = ˆ O (Γ + δ + ˜ δ, Γ + δ$_{b}$$_{1}$ e$_{b}$$_{1}$ + ˜ δ$_{b}$$_{2}$ e$_{b}$$_{2}$ ) f wsr 1 ( E ∪ ( u$_{1}$, b$_{1}$ )) ≥ ˆ O (Γ + ˜ δ, Γ + ˜ δ$_{b}$$_{2}$ e$_{b}$$_{2}$ ) + w$_{u}$$_{1}$R$_{u}$$_{1}$$_{,b}$$_{1}$δ$_{b}$$_{1}$ + w$_{u}$$_{1}$R$_{u}$$_{1}$$_{,}$$_{1}$δ. (26) can be non-monotone. Proof. The set function f $^{wsr}$( . ) in (22) defined on the family J is normalized and non-negative by construction. Due to the presence of minimum rate limits in (1) this function need not be monotone, i.e., there can exist members A , B ∈ J : A ⊆ B for which f $^{wsr}$( A ) > f $^{wsr}$( B ) . Simultaneously, there can exist members A $^{′}$, B ′ ∈ J : A ′ ⊆ B ′ for which f $^{wsr}$( A $^{′}$) ≤ f $^{wsr}$( B $^{′}$) . Then, to establish submodularity of f $^{wsr}$( . ) on the family J , it suffices to show that each f wsr m ( . ) is submodular on the family J ( m $^{)}$. Without loss of generality, we consider macro TP 1 and will prove that forall E ⊆ F ∈ J $^{(1)}$, ( u$_{1}$, b$_{1}$ ) ∈ Ω \ F : F ∪ ( u$_{1}$, b$_{1}$ ) ∈ J $^{(1)}$, Further, it suffices to prove (24) for F = E ∪ ( u$_{2}$, b$_{2}$ ) so that |F| = |E| + 1 and ( u$_{2}$, b$_{2}$ ) ∈ J $^{(1)}$. Then, we evaluate f wsr 1 ( F ∪ ( u$_{1}$, b$_{1}$ )) as described in Section III-A and in the obtained optimal allocation fractions let the share of pico TP b$_{1}$ resource assigned to user u$_{1}$ in tuple ( u$_{1}$, b$_{1}$ ) be δ$_{b}$$_{1}$ and the share of macro TP resource assigned to that user be δ . Similarly, let the share of pico TP b$_{2}$ resource assigned to user u$_{2}$ in tuple ( u$_{2}$, b$_{2}$ ) be ˜ δ$_{b}$$_{2}$ and the share of macro TP resource assigned to that user be ˜ δ . Define Γ = 1 − δ − ˜ δ and Γ = 1 − δ$_{b}$$_{1}$ e$_{b}$$_{1}$ − ˜ δ$_{b}$$_{2}$ e$_{b}$$_{2}$ . Thus, we have that where ˆ O (Γ , Γ ) is evaluated for the tuples in E under the budgets Γ and Γ . Further, we can readily verify the relations in (26). Using (25) and (26) in (24), it is now seen that a sufficient condition for (24) to hold is for (15) to be true. The latter is assured by Proposition 4. We propose Algorithm II referred to as the Greedy plus Enhanced Local Search (GELS) algorithm to optimize (1). This algorithm is an adaptation (with a slight variation) of the one proposed in [23] for non-monotone submodular set function maximization under a matroid f wsr 1 ( E ∪ ( u$_{1}$, b$_{1}$ )) − f wsr 1 ( E ) ≥ f wsr 1 ( F ∪ ( u$_{1}$, b$_{1}$ )) − f wsr 1 ( F ) . (24) f wsr 1 ( F ∪ ( u$_{1}$, b$_{1}$ )) = ˆ O (Γ , Γ ) + w$_{u}$$_{1}$R$_{u}$$_{1}$$_{,b}$$_{1}$δ$_{b}$$_{1}$ + w$_{u}$$_{1}$R$_{u}$$_{1}$$_{,}$$_{1}$δ + w$_{u}$$_{2}$R$_{u}$$_{2}$$_{,b}$$_{2}$ ˜ δ$_{b}$$_{2}$ + w$_{u}$$_{2}$R$_{u}$$_{2}$$_{,}$$_{1}$ ˜ δ, (25)