Device-to-device (D2D) communication is a promising technique to enhance the spectrum efficiency of a cellular network. However, one of the major challenges in D2D communications is to select appropriate communication modes for all possible D2D pairs in a D2D network. In this paper, we study the mode selection problems in a multi-mode and multi-pair D2D network, where the eNB can assign one of the three D2D communication modes (including local route mode, direct D2D mode, and relay D2D mode) to each D2D pair. To achieve the maximal cell-wise capacity, an optimal mode selection algorithm is proposed in this paper with the aim to share cellular users' uplink channels. The optimal mode selection problem can be modeled by an integer programming problem. Particularly, in a fully loaded cell, we discover that the optimal mode selection problem can be solved using a linear programming algorithm with a significantly reduced complexity.
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