Network Simplification in Half-Duplex: Building on Submodularity

Given an N -relay Gaussian Half-Duplex (HD) diamond network, can we achieve a significant fraction of the capacity of the full network by operating only a subset of k relays? This paper seeks to answer this question, which is also known as the network simplification problem. Building upon the recent result of Cardone et al., this work proves that there always exists a subnetwork of k = N − 1 relays that approximately (i.e., up to a constant gap) achieves a fraction N−1 N of the capacity of the full network. This fraction guarantee is shown to be tight, i.e., there exists a class of Gaussian HD diamond networks with N relays where the best (i.e., the one with the largest capacity) subnetwork of k = N − 1 relays has an approximate capacity of N−1 N of the capacity of the full network. Moreover, it is shown that any optimal schedule of the full network can be used by at least one of the ( N k ) subnetworks of k relays to achieve a worst performance guarantee of k N . The key step of the proof lies in the derivation of properties of submodular functions, which provide a combinatorial handle of the network simplification problem in Gaussian HD diamond networks.