the other hand, if ˆ λ ∈ [( m − 1) µ$_{m}$$_{−}$$_{1}$$_{,b}$, ( m − 1) µ$_{m,b}$ ] for some m = 2 , · · · , N$_{b}$ + 1 , the optimal solution comprises of assigning an identical resource share γ$_{k,b}$ = 1 / ( m − 1) with θ$_{k,}$$_{1}$ = 0 for all users k ∈ U ( b ) : R$_{k,}$$_{1}$ R$_{k,b}$ < µ$_{m,b}$ , whereas all users k ∈ U ( b ) : R$_{k,}$$_{1}$ R$_{k,b}$ ≥ µ$_{m,b}$ are assigned θ$_{k,}$$_{1}$ = 1 / ˆ λ with γ$_{k,b}$ = 0 . We next introduce another useful result that will be invoked to establish the performance guarantee of an algorithm proposed later for (2) in the sequel. Towards that end, we introduce the problem in (21) where we recall that { x$_{u,b}$ }$_{u}$$_{∈U}$ $^{′}$,b ∈B ′ 1 are given. Proposition 5. The optimal solution determined from (21) yields an objective value for (16) that is no less than the optimal objective value of (16) minus min {|B ′ $_{1}$| , |U $^{′}$| } ln(2) . Proof. We first note that the problem in (21) is indeed equivalent to (16) with the additional restriction that θ$_{u,}$$_{1}$γ$_{u,b}$ = 0 , ∀ u ∈ U ( b $^{)}$, b ∈ B ′ $_{1}$. Further invoking the result in Corollary 3 that in an optimal solution of (16), for each TP b ∈ B ′ 1 at-most one user in U ( b ) is assigned resources by both the macro TP and pico TP b . The remaining users in U ( b ) are all assigned resource by either the macro TP or by pico TP b . Suppose that user is ˆ u$_{b}$ which is assigned resource ˆ θ$_{ˆ}$ u$_{b}$, 1 by the macro and resource ˆ γ$_{ˆ}$ u$_{b}$,b by pico TP b . Then, for that user we can bound the rate as Considering all TPs in B ′ 1 we get the desired result. We are now ready to propose approximation algorithms for the problems in (1) and (2). We begin with the WSR maximization problem in (1). Let us define a ground set Ω = { ( u, b ) , u ∈ U , b ∈ P} where ( u, b ) conveys the association of user u with pico TP b . The tuple also IV. APPROXIMATION ALGORITHMS ln $^{(}$R$_{ˆ}$ u$_{b}$, $_{1}$ˆ θ$_{ˆ}$ u$_{b}$, 1 + R$_{ˆ}$ u$_{b}$$_{,b}$ˆ γ$_{ˆ}$ u$_{b}$,b ) ≤ ln(2) + max $^{{}$ln $^{(}$R$_{ˆ}$ u$_{b}$, $_{1}$ˆ θ$_{ˆ}$ u$_{b}$, 1 $^{)}$, ln ( R$_{ˆ}$ u$_{b}$$_{,b}$ˆ γ$_{ˆ}$ u$_{b}$$_{,b}$) $^{}}$, max ˜ z$_{u,}$$_{1}$ ∈{ 0 , 1 } ∀ u ∈U ′    ∑ u ∈U ′  $_{}$˜ z$_{u,}$$_{1}$ ln ( R$_{u,}$$_{1}$ ) + ∑ b ∈B ′ 1 (1 − ˜ z$_{u,}$$_{1}$ ) x$_{u,b}$ ln ( R$_{u,b}$ )$^{}$ $_{}$−   ( ∑ k ∈U ′ ˜ z$_{k,}$$_{1}$$^{)}$ ln$^{(}$ ∑ k ∈U ′ ˜ z$_{k,}$$_{1}$$^{)}$ + ∑ b ∈B ′ 1 ( ∑ k ∈U ′ (1 − ˜ z$_{k,}$$_{1}$ ) x$_{k,b}$$^{)}$ ln$^{(}$ ∑ k ∈U ′ (1 − ˜ z$_{k,}$$_{1}$ ) x$_{k,b}$$^{)}$$^{}$     (21)