  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.
