Colour-and-Forward: Relaying "what the destination needs" in the zero-error primitive relay channel

Zero-error communication over a primitive relay channel is for the first time proposed and studied. This model is used to highlight how one may exploit the channel structure to design a relaying strategy that explicitly provides “what destination needs”. We propose the Colour-and-Forward relaying scheme which constructs a graph GR of relay outputs based on the joint conditional distribution of the relay and destination outputs given the channel input. The colours of this graph GR are sent over the out-of-band link in the primitive relay channel and are shown to be information lossless in the zero-error sense; they result in the same confusability graph as if the destination had the relay’s received signal. This allows us to obtain an upper bound on the minimum required conference rate required for the relay and destination terminals to be effectively fully cooperative for any number of channel use n. It also leads to an achievable zero-error communication rate for the primitive relay channel, which may be shown to be capacity for a class of channels. I. BACKGROUND AND MOTIVATION Motivation. The core function of a relay is to help the destination in disambiguating the inputs, i.e. to provide “what the destination needs”. A relay’s goal is not to decode the message this is why Decode-and-Forward fails in general; it is not to provide “what the destination does not want”, i.e. the noise, this is why Amplify-and-Forward fails in general; nor is it desirable to waste its communication to send “what destination already possesses”. One might argue that Partial Decode-and-Forward and Compress-andForward embody the idea of providing “what the destination needs” to some extent. However, we are not aware of any explicit attempt to characterize and quantify this intuition, which could potentially lead to a new relaying strategy with improved rates. In this paper, we attempt to quantify intuition about relaying “what the destination needs” in the context of communicating over a primitive relay channel (PRC) without error, because 1) PRC is the simplest [2] network that contains a relay and 2) the imposition of zero-error constraint turns the problem into a combinatorial one. We believe this makes it easier to formalize and hope that insights may be borrowed to inspire new relaying strategies for a vanishing probability of error. Related work. Zero-error communication over a primitive relay channel at first glance seems to be a combination of two notoriously difficult and open communication problems in information theory: computing the zero-error capacity over 1The optimality of this upper bound is proved in paper [1]. a point-to-point channel2 , and the small-error capacity of a relay channel, whose capacity is unknown in general. Communication allowing a vanishing probability of error is called small-error or −error communication, while communication without error is called zero-error or 0-error communication. The small-error capacity and the zero-error capacity of a point-to-point discrete memoryless channel were both initially studied by Claude E. Shannon, in [4] in 1948 and in [5] in 1956. The zero-error capacity of a point-to-point channel (X , p(y|x),Y) with discrete finite channel input and output alphabets is characterized as the limit as the number of channel uses n → ∞ of the normalized independence number α(GX|Y ) of the n-fold AND product of the confusability graph GX|Y associated with p(y|x). This generally uncomputable limiting expression is rather unsatisfying. Even for small alphabet sizes, this is a challenging problem: Shannon’s conjecture that the capacity of the famous “pentagon graph” channel is 12 log 5 was only formally proven by Lovasz [6] 23 years later by proposing a computable-in-polynomial-time upper bound for the independence number of a graph. Thus, a computable expression for the zero-error capacity for even the simplest, point-to-point channel remains open, except for a small class of channels with perfect graphs3[3]. The primitive relay channel (PRC) proposed in [2] is a three-node relay channel introduced to decouple the multiple access and broadcast components of the standard relay channel by having the link from the relay to the destination be out of band and of fixed capacity C0, as shown in Figure 1. In paper [2], an intensive case study of the smallerror communication over a PRC is provided and it is shown that the classical relaying strategies Amplify-andForward, Decode-and-Forward and Compress-and-Forward are optimal in some classes of channels, but sub-optimal in general. We first define the new problem of zero-error communication over a primitive relay channel in Section II and state our two main questions in Section III. In Section IV, we present our main results: the construction of a new Colourand-Forward relaying scheme based on a novel compression graph GR. This scheme is “information-lossless”: together with the observation at the destination, the colour sent by the relay yields the same confusability graph as the original 2Multi-letter capacity expressions are available, but these are not generally computable except for a small class of channels with perfect graphs [3]. 3A perfect graph is a graph where the chromatic number of every induced subgraph is that subgraph’s largest clique size.