On the structure of approximately optimal schedules for half-duplex diamond networks

We consider the Gaussian diamond network where a source communicates with the destination through n non-interfering half-duplex relays. The capacity of such networks, although not known exactly, can be approximated to within a constant gap that is independent of SNR of the channels. The approximation takes the form of a linear program where the optimization is on the schedule of the relaying states. It was conjectured in [3] that there always exist optimal schedules that have at most n+1 active states, instead of the possible 2ⁿ relaying states. Making novel use of submodularity properties of cut expressions appearing in the linear program, we prove the conjecture for n = 3 and show that there exist optimal schedules with at most 6, 9 and 17 active states for n = 4, 5 and 6 relay networks, respectively.