Binary Power Optimality for Two Link Full-Duplex Network Shalanika Dayarathna, Rajitha Senanayake and Jamie Evans Department of Electrical and Electronic Engineering, University of Melbourne, Australia Email: sdayarathna@student.unimelb.edu.au, rajitha.senanayake@unimelb.edu.au, jse@unimelb.edu.au Abstract—In this paper, we analyse the optimality of binary power allocation in a network that includes full-duplex commu- nication links. Considering a network with four communicating nodes, two of them operating in half-duplex mode and the other two in full-duplex mode, we prove that binary power allocation is optimum for the full-duplex nodes when maximizing the sum rate. We also prove that, for half-duplex nodes binary power allocation is not optimum in general. However, for the two special cases, 1) the low signal-to-noise-plus-interference (SINR) regime and, 2) the approximation by the arithmetic mean-geometric mean inequality, binary power allocation is optimum for the approximated sum rate even for the half-duplex nodes. We further analyse a third special case using a symmetric network for which the optimum power allocation is binary, under a sufficient condition. Numerical examples are included to illustrate the accuracy of the results. I. I NTRODUCTION Throughput optimization in wireless networks has been very important for decades. In a wireless network, if interference is treated as Gaussian noise, we can write the Shannon theoretic rate of a node as log(1 + SINR), with SINR representing the received signal-to-interference-plus-noise ratio of the given node [1]. Maximizing the sum rate, when links have maximum power constraints, is a difficult problem because of its non- linear and non-convex nature [2]. One approach to solve this problem is the use of ap- proximation techniques to develop sub-optimal algorithms through convexification and linearization [3], [4]. However, these approximations could introduce extra constraints that cause the resulting power vector to steer away from the optimum solution in certain cases. For example, in [5], [6] a high SINR approximation is used to establish convexity in the sum rate objective function, while in [4], Taylor expansion and arithmetic mean-geometric mean approximation in low SINR region is used to establish convexity in the sum rate objective function. The approximation by construction does not allow completely turning off the power of any link at any time, which causes significant sub-optimality in the resulting power vector in certain cases. Another common approach is the use of simple power allocation structures such as the binary power allocation. “Binary” here means that a link is either “on” or “off”, with the transmitting node either operating at zero power, or maximum power, without taking any value in the continuum of possible values between zero and the maximum transmit power. This work was supported by the Australian Research Council Discovery Project under Grant DP180101205 and Discovery Early Career Researcher Award under Grant DE180100501. Interestingly, binary power allocation - as simple as it may sound - shows close to optimal results in certain interfering networks [4], [7]–[9]. More specifically, in [2], the authors consider the uplink of a single-cell where multiple users transmit to a single base station, and show, via the theory of majorization, that optimal power control is binary. In [9], the authors consider a symmetric interfering network where all direct links have one particular gain, and all the cross- links have another particular gain and show that binary power allocation is optimal when such total symmetry is maintained. The work mentioned above considers half-duplex networks where nodes cannot transmit and receive simultaneously using the same time/frequency resources. With the introduction of full-duplex mode, nodes are now allowed to transmit and re- ceive simultaneously using the same time/frequency resources. In this paper, we focus on the optimality of binary power allocation in an interference network with full-duplex links when the objective is to maximize the sum rate. Binary power allocation in full-duplex networks has re- ceived very little attention in the past [10]–[12]. In [10], [11] the authors consider a single-cell network with a full-duplex base station and two half-duplex users. By decomposing the network into two half-duplex links they show that binary power allocation is optimal when the objective is to maximize the sum rate. In [12], the authors consider a simple network with just two nodes operating in full-duplex mode and obtain a similar conclusion. They further generalize it to a multi- user network even for which the optimality of binary power allocation holds. In the present paper we take a step further, and consider a network with four nodes where two operate in half-duplex mode while the other two operate in full-duplex mode. Such a network model can practically arise in device-to-device (D2D) communication system underlaying cellular communication [13]. Within this setting, we show that binary power allocation is optimal for the full-duplex nodes but it is not optimal for the half-duplex nodes in general. We analyse the binary power optimality of the half-duplex nodes under three special cases. Firstly, the low SINR regime where the approximation log 2 (1+ SINR) ≈ SINR ln2 holds. Secondly, using the approxi- mation by the arithmetic mean-geometric mean inequality and thirdly using a symmetric network for which binary power allocation is optimum under a sufficient condition. We include numerical results to further illustrate our results.