Network Simplification in Half-Duplex: Building on Submodularity
  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. 
Martina Cardone, Yahya H. Ezzeldin, Christina Fragouli, Daniela   Tuninetti
cs.IT math.IT