On the Optimality of Simple Schedules for Networks with Multiple Half-Duplex Relays

This paper studies networks that consist of N half-duplex relays assisting the communication between a source and a destination. In ISIT'12 Brahma et al. conjectured that in Gaussian half-duplex diamond networks (i.e., without a direct link between the source and the destination, and with N non-interfering relays), an approximately optimal relay scheduling policy (i.e., achieving the cut-set upper bound to within a constant gap uniformly over all channel gains) has at most N+1 active states (i.e., at most N+1 out of the 2^N possible relay listen-Transmit configurations have a strictly positive probability). Such relay scheduling policies were referred to as simple. In ITW'13, we conjectured that simple approximately optimal relay scheduling policies exist for any Gaussian half-duplex multi-relay network irrespectively of the topology. This paper formally proves this more general version of the conjecture and shows it holds beyond Gaussian noise networks. In particular, for any class of memoryless half-duplex N-relay networks with independent noises and for which independent inputs are approximately optimal in the cut-set upper bound, an approximately optimal simple relay scheduling policy exists. The key step of the proof is to write the minimum of the submodular cut-set function by means of its Lovász extension and use the greedy algorithm for submodular polyhedra to highlight structural properties of the optimal solution. This, together with the saddle-point property of min-max problems and the existence of optimal basic feasible solutions for linear programs, proves the conjecture. As an example, for N-relay Gaussian networks with independent noises, where each node is equipped with multiple antennas and where each antenna can be configured to listen or transmit irrespectively of the others, the existence of an approximately optimal simple relay scheduling policy with at most N+1 active states, irrespectively of the total number of antennas in the system, is proved.