半双工网络简化：基于子模块化
Network Simplification in Half-Duplex: Building on Submodularity
摘要：
本文探讨了高斯半双工（HD）菱形网络的背景下的{ \it网络简化}问题。具体地，给定$N$ -中继菱形网络，该问题寻求导出关于最佳$k$ -中继子网络的容量的基本保证，作为全网络容量的函数。这项工作的主要重点是当$k=N-1$继电器被选择出的$N$可能的情况下。首先，简单算法，其移除具有最小容量的中继（即，最差中继），并且示出了剩余的$(N-1)$ -中继子网具有近似（即，最佳直到恒定间隙）HD容量，其至少是整个网络的近似HD容量的一半。该分数保证被证明是紧的，如果只有单个中继容量是已知的，即，存在一类具有$N$中继的高斯HD菱形网络，其中通过去除最差的中继，剩余的$k=N-1$中继的子网具有等于全网络的近似容量的一半的近似容量。接下来，这项工作证明了一个基本的保证，这改善了前一部分：总是存在一个子网络的$k=N-1$继电器，实现至少一部分$\frac{N-1}{N}$的近似容量的完整的网络。这部分被证明是紧的，它表明，任何最佳的时间表的整个网络可以使用的至少一个$N$子网络的$N-1$继电器，以实现最坏情况下的性能保证$\frac{N-1}{N}$。此外，这些结果被扩展到一般$k \in [1:N]$的分数保证的下界。证明的关键步骤在于子模函数性质的推导，这为高斯HD菱形网络中的网络简化问题提供了一个组合处理方法。
英文摘要：
This paper explores the {\it network simplification} problem in the context of Gaussian Half-Duplex (HD) diamond networks. Specifically, given an $N$-relay diamond network, this problem seeks to derive fundamental guarantees on the capacity of the best $k$-relay subnetwork, as a function of the full network capacity. The main focus of this work is on the case when $k=N-1$ relays are selected out of the $N$ possible ones. First, a simple algorithm, which removes the relay with the minimum capacity (i.e., the worst relay), is analyzed and it is shown that the remaining $(N-1)$-relay subnetwork has an approximate (i.e., optimal up to a constant gap) HD capacity that is at least half of the approximate HD capacity of the full network. This fraction guarantee is shown to be tight if only the single relay capacities are known, i.e., there exists a class of Gaussian HD diamond networks with $N$ relays where, by removing the worst relay, the subnetwork of the remaining $k=N-1$ relays has an approximate capacity equal to half of the approximate capacity of the full network. Next, this work proves a fundamental guarantee, which improves over the previous fraction: there always exists a subnetwork of $k=N-1$ relays that achieves at least a fraction $\frac{N-1}{N}$ of the approximate capacity of the full network. This fraction is proved to be tight and it is shown that any optimal schedule of the full network can be used by at least one of the $N$ subnetworks of $N-1$ relays to achieve a worst-case performance guarantee of $\frac{N-1}{N}$. Additionally, these results are extended to derive lower bounds on the fraction guarantee for general $k \in [1:N]$. The key steps in the proofs lie in the derivation of properties of submodular functions, which provide a combinatorial handle on the network simplification problem in Gaussian HD diamond networks.
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