This paper considers a single-source single-destination half-duplex $n$ -relay network with arbitrary topology, where the source communicates with the destination through a direct link and with the help of $n$ half-duplex relays. The focus is on the linear deterministic approximation of the Gaussian noise network model. First, sufficient conditions under which operating the network in the $n+1$ energy-efficient states (out of the $2^{ n}$ possible states) is sufficient to achieve the approximate capacity (that is, an additive gap approximation of the Shannon capacity) are characterized. Specifically, these $n+ 1$ energy-efficient states are those in which at most one relay is in transmit mode while the rest of the relays are in receive mode. Under such sufficient network conditions, closed-form expressions for the scheduling and the approximate capacity are provided. Then, a time-block relaying scheme, where at each point in time at most one relay is in transmit mode, is designed. In particular, the designed relaying scheme leverages information flow preservation at each relay to explicitly provide the information that each relay is exclusively responsible to store and forward to the destination. Furthermore, the destination can decode the information bits sent by the source in block $B$ by the end of block $B+1$ , and the proposed scheme is shown to achieve the approximate capacity whenever the sufficient conditions are satisfied. Such features make the designed scheme relevant for practical use.
