This paper investigates the problem of characterizing the capacity of Half-Duplex (HD) line networks, where a source node communicates to a destination node through a multi-hop path of $N$ relays. If the relays operate in Full-Duplex (FD), it is well known that the capacity of the line network equals the minimum among the point-to-point link capacities in the path. In contrast, this paper considers a different case where the relays operate in HD. In the first part of the paper, it is shown that the approximate capacity (optimal up to a constant additive gap that only depends on the number of nodes in the network) of an HD $N$ -relay line network equals half the minimum of the harmonic means of the point-to-point link capacities of each two consecutive links in the path. It is then proved that the $N+1$ listen/transmit states (out of the $2^{N}$ possible ones) sufficient to characterize the approximate capacity can be found in linear time. In the second part of the paper, it is shown that the problem of finding the path that has the largest HD approximate capacity in a network that can be represented as a graph is NP-hard. However, if the number of cycles in the network is polynomial in the number of nodes, then a polynomial-time algorithm can indeed be designed.

中文翻译：

---

半双工线路网络的近似容量

本文研究了表征半双工 (HD) 容量的问题 线网，其中源节点通过多跳路径与目的节点通信 $N$ 继电器。如果中继以全双工 (FD) 运行，众所周知，线路网络的容量等于路径中点对点链路容量中的最小值。相比之下，本文考虑了继电器在高清模式下运行的不同情况。在论文的第一部分中，显示了 HD 的近似容量（最佳达到仅取决于网络中节点数的恒定附加间隙） $N$ -中继线网络等于路径中每两个连续链路的点对点链路容量的调和平均值的最小值的一半。然后证明 $N+1$ 侦听/传输状态（在 $2^{N}$ 可能的）足以表征近似容量可以在线性时间内找到。在论文的第二部分中，表明在可以表示为图的网络中找到具有最大 HD 近似容量的路径的问题是 NP-hard 问题。但是，如果网络中的循环数是节点数的多项式，那么确实可以设计多项式时间算法。