Wireless network simplification: The Gaussian N-relay diamond network

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 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 N-relay diamond networks, where every subset of k relays can provide at most k/k+1 of the total capacity. The approximation is within 3 logN + 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 1/k+1 and additive gap of 3 logN + 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 approach where multiplicative and additive gaps are allowed simultaneously and are traded through an auxiliary parameter.