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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. [ABSTRACT FROM PUBLISHER]

This paper considers a heavy loaded full-duplex (FD) small cell consisting of one FD base station and multiple half-duplex users where the number of subchannels is less than the number of uplink and downlink users. With the objective of maximizing the total sum-rate, a joint power allocation, and user pairing problem is formulated. To solve this problem, a novel decomposition method is proposed, which decouples the power allocation and user pairing problem into two subproblems. It is proven that the optimal solution of subproblems is the optimal solution of the original problem. Then, optimal algorithms are developed for the two subproblems. Numerical results show that the proposed algorithm achieves better performance compared with the baselines of the greedy algorithm and random pairing algorithm. [ABSTRACT FROM AUTHOR]