We consider the Gaussian $N$-relay diamond network, where a source wants to
communicate to a destination node through a layer of $N$-relay nodes. We
investigate the following question: What fraction of the capacity can we
maintain by using only $k$ out of the $N$ available relays? We show that in
every Gaussian $N$-relay diamond network, there exists a subset of $k$ relays
which alone provide approximately a fraction $\frac{k}{k+1}$ of the total
capacity. The result holds independent of the number of available relay nodes
$N$, the channel configurations and the operating SNR. The result is tight
in the sense that there exists channel configurations for $k+1$-relay diamond
networks, where every subset of $k$ relays can provide at most a fraction of 
$\frac{k}{k+1}$ of the total capacity. The approximation is within 
$3\log N+3k$ bits/s/Hz to the capacity. 

This result also provides a new approximation to the capacity of the Gaussian
$N$-relay diamond network which is up to a multiplicative gap of 
$\frac{k+1}{k}$ and additive gap of $3\log N+3k$. The current approximation
results in the literature either aim to characterize the capacity within an
additive gap by allowing no multiplicative gap or vice a versa. Our result
suggests a new approximation philosophy where multiplicative and additive gaps
are allowed simultaneously and are traded through an auxiliary parameter.