We formulate the joint power and channel allocation problem (JPCAP) for device to device (D2D) communication as a cost minimization problem, where cost is defined as a linear combination of the number of channels used and total power requirement. We first show that JPCAP is a NP-hard problem and providing $n^{1∕ϵ}$ approximation for JPCAP $\forall ϵ>0$ is also NP-hard. Then we propose a mixed integer linear programming (MILP) to solve this problem. As solving MILP is a NP-hard problem we propose a greedy channel and power allocation (GCPA) algorithm to assign channels and powers to the links. We design GCPA in such a fashion that there exists an order of the links for which it produces optimum solution. We show that an order is equivalent to many orders and hence design an incremental algorithm (IA) to efficiently search good orders. Finally using IA we develop a randomized joint channel and power allocation (RJCPA) algorithm. We show that if a certain condition holds we can find the optimum in expected polynomial time else a slowly growing exponential time with very high probability. We then theoretically calculate the expected cost and energy efficiency (EE) produced by RJCPA. Through simulation, we show that RJCPA outperforms two existing approaches with respect to both cost and EE significantly. Finally we validate our theoretical findings through simulation.

我们将设备到设备 (D2D) 通信的联合功率和信道分配问题 (JPCAP) 公式化为成本最小化问题，其中成本被定义为使用的信道数量和总功率需求的线性组合。我们首先证明 JPCAP 是一个 NP-hard 问题并提供$n^{1∕\epsilon }$ JPCAP 的近似值 $\forall \epsilon >0$也是 NP-hard。然后我们提出了一个混合整数线性规划（MILP）来解决这个问题。由于解决 MILP 是一个 NP-hard 问题，我们提出了一种贪婪信道和功率分配 (GCPA) 算法来为链路分配信道和功率。我们以这样一种方式设计 GCPA，即存在一个链接顺序，可以为其生成最佳解决方案。我们表明一个订单相当于许多订单，因此设计了一种增量算法 (IA) 来有效地搜索好的订单。最后使用 IA，我们开发了一种随机联合信道和功率分配 (RJCPA) 算法。我们表明，如果某个条件成立，我们可以在预期多项式时间内找到最优值，否则以非常高的概率缓慢增长的指数时间。然后我们从理论上计算 RJCPA 产生的预期成本和能源效率 (EE)。通过模拟，我们表明 RJCPA 在成本和 EE 方面均显着优于两种现有方法。最后，我们通过模拟验证了我们的理论发现。