max x$_{u,b}$ ∈{ 0 , 1 } ∀ u ∈U ,b ∈S { ∑ u ∈U ∑ b ∈S x$_{u,b}$ ln ( R$_{u,b}$ ) − ∑ b ∈S ( ∑ k ∈U x$_{k,b}$$^{)}$ ln$^{(}$ ∑ k ∈U x$_{k,b}$$^{)}}$ s . t . ∑ b ∈S x$_{u,b}$ = 1 , ∀ u ∈ U . (28) node b ∈ B$_{m}$ for any m ∈ M are also connected to the macro m . Further, each user associated to a macro TP is also associated to a pico TP in the set of pico TPs assigned to that macro. Then, the allocation fractions are optimized as described in Section III-B. The performance guarantee of Algorithm III is established below, where we let Π ≥ 0 denote the (additive) guarantee pertaining to the approach used to solve (28), i.e., the value yielded by the obtained output is no less than the corresponding optimal objective value of (28) minus Π (so that Π = 0 for the optimal algorithm [5]). The key insight used in the proof is from Corollary 3 that for any valid association, considering each macro and each pico assigned to that macro, optimal allocation fractions entail only one user receiving a positive resource share from both that macro and pico. Theorem 4. Algorithm III provides an output that yields an objective value for (2) that is no less than the optimal objective value of (2) minus Π + min { K, ∑ m ∈M |B$_{m}$|} ln(2) . Proof. We first note that (28) is equivalent to (2) with the additional constraint that γ$_{u,b}$θ$_{u,m}$ = 0 , ∀ u ∈ U , b ∈ B$_{m}$ , m ∈ M . Suppose that (28) is solved using an approach that offers a guarantee of Π . 2 Then, note that the obtained solution feasible for (2) and invoking Proposition 5 (once for each macro TP together with its assigned set of pico TPs and the users associated to them) we can conclude that the attained objective value is no less than the optimal one minus the claimed additive factor. The remaining steps of Algorithm III further improve the solution at hand and hence further reduce the gap to optimal, which proves the theorem. We remark that another way to view the performance guarantee of Algorithm III (when Π = 0 ) is as follows. Let us scale all the peak rates of any set of min { K, ∑ m ∈M |B$_{m}$|} users by 2 and obtain an output by Algorithm III. Then, the objective value in (2) yielded by the solution at hand, will be no less than the one yielded by the optimal solution using the original peak rates. $^{2}$Note that since the objective function in (28) and (2) can be negative, we can only offer additive guarantees instead of multiplicative ones.