Abstract-In this paper we study a simple question: when are dynamic relaying strategies essential in optimizing the diversitymultiplexing tradeoff (DMT) in half-duplex wireless relay networks? This is motivated by apparently two contrasting results even for a simple 3 node network, with a single half-duplex relay. When all channels in the system are assumed to be independent and identically fading, a static schedule where the relay listens half the time and transmits half the time combined with quantizemap-and-forward (QMF) relaying is known to achieve the fullduplex performance. However, when there is no direct link between the source and the destination, a dynamic decodeand-forward (DDF) strategy is needed to achieve the optimal tradeoff. In this case, a static schedule is strictly suboptimal and the optimal tradeoff is significantly worse than the fullduplex performance. In this paper we study the general case when the direct link is neither as strong as the other links nor fully non-existent, and identify regimes where dynamic schedules are necessary and those where static schedules are enough. We identify four qualitatively different regimes for the single relay channel where the tradeoff between diversity and multiplexing is significantly different. We show that in all these regimes one of the above two strategies is sufficient to achieve the optimal tradeoff by developing a new upper bound on the best achievable tradeoff under channel state information available only at the receivers. A natural next question is whether these two strategies are sufficient to achieve the DMT of more general half-duplex wireless networks with a larger number of relays. We propose a generalization of the two existing schemes through a dynamic QMF strategy, where the relay listens for a fraction of time depending on received CSI but not long enough to be able to decode. We show that such a dynamic QMF (DQMF) strategy is needed to achieve the optimal DMT in a parallel channel with two relays, outperforming both DDF and static QMF strategies.

Consider a Gaussian relay network where a source node communicates to a destination node with the help of several layers of relays. Recent work has shown that compress-andforward-based strategies can achieve the capacity of this network within an additive gap. Here, the relays quantize their received signals at the noise level and map them to random Gaussian codebooks. The resultant gap to capacity is independent of the SNRs of the channels in the network and the topology, but is linear in the total number of nodes. In this paper, we provide an improved lower bound on the rate achieved by the compress-andforward-based strategies (noisy network coding in particular) in arbitrary Gaussian relay networks, whose gap to capacity depends on the network not only through the total number of nodes but also through the degrees of freedom of the min cut of the network. We illustrate that for many networks, this refined lower bound can lead to a better approximation of the capacity. In particular, we demonstrate that it leads to a logarithmic rather than linear capacity gap in the total number of nodes for certain classes of layered networks. The improvement comes from quantizing the received signals of the relays at a resolution decreasing with the total number of nodes in the network. This suggests that the rule-of-thumb in the literature of quantizing the received signals at the noise level can be highly suboptimal.Index Terms-Relay networks, gap to capacity, noisy network coding, network topology, quantization.

The capacity regions of semideterministic multiuser channels, such as the semideterministic relay channel and the multiple access channel with partially cribbing encoders, have been characterized using the idea of partial-decode-forward. However, the requirement to explicitly decode part of the message at intermediate nodes can be restrictive in some settings; for example, when nodes have different side information regarding the state of the channel. In this paper, we generalize this scheme to $\textit{cooperative-bin-forward}$ by building on the observation that explicit recovering of part of the message is not needed to induce cooperation. Instead, encoders can bin their received signals and cooperatively forward the bin index to the decoder. The main advantage of this new scheme is illustrated by considering state-dependent extensions of the aforementioned semideterministic setups. While partial-decode-forward is not applicable in these new setups, cooperative-bin-forward continues to achieve capacity.Comment: submitted to IEEE Transactions on Information Theory, presented in part at ISIT 2015; version-2 contains an illustrative exampl

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Consider a Gaussian relay network where a number of sources communicate to a destination with the help of several layers of relays. Recent work has shown that a compress-andforward based strategy at the relays can achieve the capacity of this network within an additive gap. In this strategy, the relays quantize their observations at the noise level and map it to a random Gaussian codebook. The resultant capacity gap is independent of the SNR's of the channels in the network but linear in the total number of nodes.In this paper, we show that if the relays quantize their signals at a resolution decreasing with the number of nodes in the network, the additive gap to capacity can be made logarithmic in the number of nodes for a class of layered, time-varying wireless relay networks. This suggests that the rule-of-thumb to quantize the received signals at the noise level used for compress-andforward in the current literature can be highly suboptimal.