max γ$_{u,b}$,θ$_{u,}$$_{1}$ ∈ [0 , 1] ∀ u ∈U $^{′}$,b ∈B ′ 1    ∑ b ∈B ′ 1 ∑ u ∈U ( b ) ( w$_{u}$R$_{u,}$$_{1}$θ$_{u,}$$_{1}$ + w$_{u}$R$_{u,b}$γ$_{u,b}$ )$^{}$   s . t . ∑ u ∈U ′ θ$_{u,}$$_{1}$ ≤ Γ & ∑ u ∈U ( b ) γ$_{u,b}$ ≤ Γ$_{b}$ ∀ b ∈ B ′ $_{1}$; R$_{u}$ △ = R$_{u,}$$_{1}$θ$_{u,}$$_{1}$ + R$_{u,b}$γ$_{u,b}$ ∈ [ R min u , R max u ] , ∀ u ∈ U ( b $^{)}$, b ∈ B ′ $_{1}$; (3) III. CHARACTERIZING OPTIMAL ALLOCATION FRACTIONS A. Optimal allocation fractions of (1) We begin our quest by characterizing the optimal allocation fractions of (1) and (2) for any given user association. This then enables the design of approximation algorithms. We now proceed to characterize the solution of (1) for any given user association. We note that upon fixing the user association in (1) (i.e., upon fixing { x$_{u,b}$, z$_{u,m}$ } ) the problem in (1) decouples into |M| sub-problems, one for each macro TP. Consequently, we focus our attention on the subproblem corresponding to any macro TP, say with index 1 , and suppose that any subset of users U ′ ⊆ U is associated to that macro by the given association. Then, for each b ∈ B$_{1}$ , let U ( b ) = { u ∈ U : x$_{u,b}$ = 1 } denote the associated user set such that U ( b ) ∩ U ( b $^{′}$) = φ, b = b $^{′}$, where φ denotes the empty set and ∪$_{b}$$_{∈B}$$_{1}$U ( b ) = U $^{′}$. Let B ′ 1 denote the set of all pico TPs in B$_{1}$ with at-least one associated user. In addition, we consider a budget constraint for each pico TP, Γ$_{b}$ ∈ [0 , 1] , ∀ b ∈ B ′ 1 and another one for the macro, Γ ∈ [0 , 1] . With these in hand we pose the problem in (3), which we assume to be feasible. Note that without loss of generality we can assume that each pico TP is resource limited, i.e., ∑ u ∈U ( b ) R max u R$_{u,b}$ > Γ$_{b}$ . This is because otherwise we can simply assign maximum possible resource from TP b to each user in U ( b ) and remove those from further consideration. We will use the term slack to denote the resource assigned to a user in excess of its minimum rate requirement. For convenience, for each user k ∈ U ( b $^{)}$, b ∈ B ′ 1 we let R$_{k}$ = R$_{k,}$$_{1}$θ$_{k,}$$_{1}$ + R$_{k,b}$γ$_{k,b}$ and supress the dependence of R$_{k}$ on θ$_{k,}$$_{1}$, γ$_{k,b}$ . To analyze (3) we offer the following result that can be derived by carefully manipulating the K.K.T. conditions. Proposition 1. The following conditions must be satisfied by any optimal solution of (3):