10 be formulated for the general SINR regime as follows: minimize p$_{n,k,l}$ log$_{2}$ L ∏ l =1 K ∏ k =1 N ∏ n =1 ( σ 2 + I$_{n,l}$ p$_{n,k,l}$h$_{n,k,l}$ + σ 2 + I$_{n,l}$ ) α$_{n,k,l}$ subject to N ∑ n =1 α$_{n,k,l}$p$_{n,k,l}$ ≤ P$_{k,}$$_{max}$ , ∀ k, ∀ l (12) Note that the numerator and denominator in (12) are posynomials and minimizing a ratio between two posynomials is referred to be a truly non-convex NP hard intractable problem known as complimentary GP. However, this problem can be transformed into GP by letting the denominator f ( p ) = p$_{n,k,l}$h$_{n,k,l}$ + σ 2 + I$_{n,l}$ = ∑ L l =1 ∑ K k $_{=1}$u$_{n,k,l}$ ( p ) and approximating the denominator f ( p ) with a monomial using the arithmetic/geometric mean inequality as follows: L ∑ l =1 K ∑ k =1 u$_{n,k,l}$ ( p ) ≥ L ∏ l =1 K ∏ k =1 ( u$_{n,k,l}$ ( p ) s$_{n,k,l}$ ) s$_{n,k,l}$ (13) where s$_{n,k,l}$ = u$_{n,k,l}$ ( p$_{0}$ ) f ( p$_{0}$ ) . Thus, the problem can be solved by extending the single condensation method presented in [14] for multi-cell scenario. The details of centralized scheme A are presented in Algorithm 2. The initial allocation phase has a complexity of O ( KN $^{2}$) which is the same as Algorithm 1. Next, we perform a one dimensional search for the user in cell l with maximum incremental throughput at subcarrier n . The process is repeated for each subcarrier and cell. Thus, the com- putational complexity of this step is O ( KNL ) . Since, the process continues until convergence, (i.e., M iterations), the complexity of this step can be written as O ( KNLM ) . Finally, the total complexity of subcarrier allocation phase is O ( KN 2 + NKLM ) . The complexity of Phase II is difficult to determine, however , it can be measured in terms of the degree of difficulty (DoD) that in turn relies on the number ofconstraints and variables associated with the GP [15]. Since we are dealing with LK power constraints and LKN power variables, the total computational complexity of centralized scheme A is O ( KN $^{2}$+ NKLM ) + DoD ( LKN ) . Apparently it seems that implementing centralized GP/successive GP based schemes may not DRAFT March 19, 2018 B. Centralized Scheme A: Complexity Analysis