When an Energy-Efficient Scheduling is Optimal for Half-Duplex Relay Networks?

This paper considers a diamond network with $n$ interconnected relays, namely a network where a source communicates with a destination by hopping information through $n$ communicating/interconnected relays. Specifically, the main focus of the paper is on characterizing sufficient conditions under which the $n$ + 1 states (out of the 2n possible ones) in which at most one relay is transmitting suffice to characterize the approximate capacity, that is the Shannon capacity up to an additive gap that only depends on n. Furthermore, under these sufficient conditions, closed form expressions for the approximate capacity and scheduling (that is, the fraction of time each relay should receive and transmit) are provided. A similar result is presented for the dual case, where in each state at most one relay is in receive mode.