Cut-set bound

The cut-set bound is a result in multiterminal information theory which bounds the achievable rates of reliable communication in a memoryless network. It is an outer bound on the rate region.

= The bound =

We first consider the discrete memoryless network with a single message defined by $P(y_2, y_3, \ldots, y_N | x^N)$. The transmitter is terminal $1$ and the set of possible destinations is $\mathcal{D} = \{2, \ldots, N\}$. A source set is a subset $\mathcal{S} \subset \{1, 2, \ldots, N\}$ such that $1 \in \mathcal{S}$, and $\mathcal{S}^c$ is the complement of $\mathcal{S}$ in $\{1, 2, \ldots, N\}$.

Theorem. The capacity $C$ of $P(y_2, y_3, \ldots, y_N | x^N)$ is upper bounded: \begin{align} C \le \max_{p(x^N)} \min_{k \in \mathcal{D}} \min_{\mathrm{source\ set} \mathcal{S}} I(X(\mathcal{S}; Y(\mathcal{S}^c) ~|~ X(\mathcal{S}^c) ) \end{align}

The intuition for this bound is that the capacity between terminal 1 and any terminal in $\mathcal{D}$ is upper bounded by considering the capacity of a channel in which all of the terminals in a source set $\mathcal{S}$ cooperate to send the message and all of the terminals in $\mathcal{S}^c$ cooperate in decoding it.

Abbas El Gamal and Young-Han Kim, Lecture Notes on Network Information Theory, arXiv:1001.3404v4 [cs.IT]

A. El Gamal, “On information ﬂow in relay networks,” in Proc. IEEE National Telecom Conference, Nov. 1981, vol. 2, pp. D4.1.1–D4.1.4.