On Proportional Fairness in Power Allocation for Two-Tone Spectrum-Sharing Networks

To efficiently trade off system sum rate and link fairness, power allocation in wireless spectrum-sharing networks often concentrates upon proportional fairness. The corresponding problem has been proved to be convex for a single tone but NP-hard for more than two tones. However, in the two-tone case, the complexity of the problem for achieving proportional fairness has not been addressed yet. In this paper, we prove that the issue of proportional-fairness optimization for the two-tone situation is NP-hard by reducing the problem of finding the maximum independent set in an undirected graph to it. Moreover, a computationally efficient algorithm is proposed to provide an efficient suboptimal solution. Simulation results are presented to illustrate the effectiveness of our proposal.