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

Consider a multi-tone multi-user communication system with K interfering users and N 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 (N=2) 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 N. 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.