  This paper investigates the simplification problem in Gaussian Half-Duplex
(HD) diamond networks. The goal is to answer the following question: what is
the minimum (worst-case) fraction of the total HD capacity that one can always
achieve by smartly selecting a subset of $k$ relays, out of the $N$ possible
ones? We make progress on this problem for $k=1$ and $k=2$ and show that for
$N=k+1, \ k \in \{1,2\}$ at least $\frac{k}{k+1}$ of the total HD capacity is
always {approximately (i.e., up to a constant gap)} achieved. Interestingly,
and differently from the Full-Duplex (FD) case, the ratio in HD depends on $N$,
and decreases as $N$ increases. For all values of $N$ and $k$ for which we
derive worst case fractions, we also show these to be {approximately} tight.
This is accomplished by presenting $N$-relay Gaussian HD diamond networks for
which the best $k$-relay subnetwork has {an approximate} HD capacity equal to
the worst-case fraction of the total {approximate} HD capacity. Moreover, we
provide additional comparisons between the performance of this simplification
problem for HD and FD networks, which highlight their different natures.
