We study the half duplex multiple-relay channel (MRC) where every node can either
transmit or listen but not both at the same time. We derive a capacity upper bound based
on a max-flow min-cut argument and achievable transmission rates based on the decode-
forward coding strategy (DF), for both the half duplex discrete memoryless MRC and the
half duplex phase fading Gaussian MRC. The upper bound and achievable rates are
functions of the transmit state vector (a description of which nodes transmit and which
receive). More precisely, they are functions of the time fraction of different transmit state
vectors, which we term a schedule.
We formulate the optimal scheduling problem as a max-min optimization to find the
schedule that maximizes the DF rate for the half duplex MRC. We use a technique based on
minimax hypothesis testing to solve this problem and demonstrate it on a four-node MRC,
getting closed form solutions in certain scenarios. For the phase fading Gaussian channel,
surprisingly, we discover that optimal schedules can be solved using linear programming.