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We study the half-duplex multiple-relay channel (HD-MRC) where every node can either transmit or listen but cannot do both at the same time. We obtain a capacity upper bound based on a max-flow min-cut argument and achievable transmission rates based on the decode-forward (DF) coding strategy, for both the discrete memoryless HD-MRC and the phase-fading HD-MRC. We discover that both the upper bound and the achievable rates are functions of the transmit/listen state (a description of which nodes transmit and which receive). More precisely, they are functions of the time fraction of the different states, which we term a schedule. We formulate the optimal scheduling problem to find an optimal schedule that maximizes the DF rate. The optimal scheduling problem turns out to be a maximin optimization, for which we propose an algorithmic solution. We demonstrate our approach on a four-node multiple-relay channel, obtaining closed-form solutions in certain scenarios. Furthermore, we show that for the received signal-to-noise ratio degraded phase-fading HD-MRC, the optimal scheduling problem can be simplified to a max optimization.