On the complexity of maximizing the minimum Shannon capacity in Wireless Networks by joint channel assignment and power allocation

We consider wireless telecommunications systems with orthogonal frequency bands, where each band is referred to as a channel, e.g., orthogonal frequency-division multiple access (OFDMA). For a given snap-shot in time, two joint channel assignment and power allocation optimization problems are presented, one in downlink and one in uplink. The objective is to maximize the minimum total Shannon capacity of any mobile user in the system, subject to system constraints. The corresponding decision problems are proved to be NP-hard. We also show that for any constant ρ > 0, a sufficiently large number of channels ensure that the optimization problems are not ρ-approximable, unless P is equal to NP.