On the Complexity of Optimal Power Allocation in a Multi-Tone Multiuser Communication System

Consider a multi-tone multi-user communication system with <inline-formula> <tex-math notation="LaTeX">$\mathbf {K}$ </tex-math></inline-formula> interfering users and <inline-formula> <tex-math notation="LaTeX">$\mathbf {N}$ </tex-math></inline-formula> available tones. An effective approach to mitigate interference is through power control at transmitters. In this paper, we consider optimal power allocation to maximize a system utility function, and show that for the two tone cases (<inline-formula> <tex-math notation="LaTeX">$\mathbf {N{=}2}$ </tex-math></inline-formula>) with min-rate, harmonic mean, and geometric mean utility functions, the corresponding optimal power allocation problem is NP-hard. This result fills an important gap in the existing literature, which settled the complexity status of different cases involving various utility functions and values of <inline-formula> <tex-math notation="LaTeX">$\mathbf {N}$ </tex-math></inline-formula>. Our proof is through a reduction from the partitioning problem for the min-rate utility function, and from the independent set problem for the harmonic mean and geometric mean utility functions.