1: Initialize with MaxIter ≥ 1 , ˜ Ω = Ω , Δ > 0 and ˆ G = φ . 2: Repeat %Optional Greedy Stage 3: Determine ( k $^{′}$, b$^{′}$) as the tuple in ˜ Ω \ ˆ G which satisfies ˆ G ∪ ( k $^{′}$, b$^{′}$) ∈ J and offers the best gain f $^{wsr}$( ˆ G ∪ ( k $^{′}$, b$^{′}$)) − f $^{wsr}$( ˆ G ) , among all such tuples. 4: If the best gain is positive update ˆ G = ˆ G ∪ ( k $^{′}$, b$^{′}$) . 5: Until the best gain is not positive. 6: Set ˘ G = ˆ G , Iter = 0 . 7: Repeat %Enhanced Local Search (LS) Stage 8: Increment Iter = Iter + 1 . 9: Swap: Find a pair of tuples: ( k $^{′}$, b$^{′}$) ∈ ˘ G and ( k, b ) ∈ ˜ Ω \ ˘ G such that the swapping ( k $^{′}$, b$^{′}$) ∈ ˘ G with ( k, b ) is in J and results in the best gain among such swaps. 10: Deletion: Find a tuple: ( k $^{′}$, b$^{′}$) ∈ ˘ G such that deleting this tuple results in the best gain among such deletions. 11: Addition: Find a tuple: ( k $^{′}$, b$^{′}$) ∈ ˜ Ω \ ˘ G such that ˘ G ∪ ( k $^{′}$, b$^{′}$) ∈ J and adding this tuple results in the best gain among such additions. 12: Determine the overall best among the three best gains and compute threshold Δ f $^{wsr}$( ˘ G ) 13: If the overall best gain is at-least as as large as the threshold, then update ˘ G according to the overall best choice. 14: Until the overall best gain is less than threshold or Iter = MaxIter . 15: Repeat Greedy and Enhanced LS stages intialized with ˜ Ω = Ω \ ˘ G and let ˜ G be the obtained output. 16: Output the choice yielding the larger of { f $^{wsr}$( ˘ G ) , f $^{wsr}$( ˜ G ) } . 

1: Initialize with U , M , B$_{m}$ , ∀ m ∈ M . 2: Set S = M ∪ ( ∪$_{m}$$_{∈M}$B$_{m}$ ) and determine user associations { x$_{u,b}$ } ,u ∈ U ,b ∈ S by solving (28) 3: For each macro TP m ∈ M Do 4: Consider each user with x$_{u,m}$ = 1 and associate that user with the pico TP in B$_{m}$ yielding the strongest received power for that user. 5: Using the obtained association for TPs in B$_{m}$ obtain the optimal allocation fractions using Theorem 1. 6: End For 7: Output the user associations and allocation fractions. 

## V. SIMULATION RESULTS 

We now present our simulation results obtained for an LTE HetNet deployment. We emulate a HetNet comprising of 57 macro cells with 10 pico cells being assigned to each macro and with a full buffer traffic model. Each macro base-station transmits with a power of 46 dBm whereas the transmit power at each pico node is 40 dBm and the system bandwidth is 10 MHz. A noise PSD of − 174 dB/Hz with a noise figure of 9 dB were assumed. Other major parameters such as the distributions used to drop users, macro and pico nodes are all as per 3GPP guidelines. 

