Cut-set bounds

Cut-set bounds are the simplest fundamental outer bounds for capacity regions of networks.

Discrete memoryless network
Consider a network with $n$ nodes $u_1,u_2,\ldots, u_n.$ Let the transmission of node $u_i$ be $x_i$ and its reception be $y_i.$ Suppose the reception $(y_1,y_2,\ldots, y_n)$ is related to the input $(x_1,x_2,\ldots,x_n)$ by a memoryless transition function $p(y_1,y_2,\ldots, y_n|x_1,x_2,\ldots, x_n).$ Suppose each node $i$ wishes to communicate to node $j$ at rate $R_{i,j}$ and the messages across node pairs are all independent.

The cutset bound provides an outer bound to the set of achievable rate-tuples $\{R_{i,j}:1\leq i,j\leq n, i\neq j\}.$

Consider a subset $S\subseteq\{1,2,\ldots, n\}.$ The cut-set bound gives the inequality:

$$\sum_{i\in S, j\in S^c} R_{i,j}\leq \sup_{p(x_1,x_2,\ldots, x_n)}I(X_S;Y_{S^c}|X_{S^c})$$

Wireline networks
For the special case of network coding, the cut-set bound simplifies to a sum of capacities of links.

As an example, consider the butterfly network alongside - source $s_1$ communicating to destination $t_1$ at rate $R_1$ and source $s_2$ communicating to destination $t_2$ at rate $R_2.$

The choice of $S=\{s_1,t_2,u\}$ gives the bound $R_1\leq C_a.$

The choice of $S=\{s_2,t_1,u\}$ gives the bound $R_2\leq C_a.$

Further, the choice of $S=\{s_1,s_2,u\}$ gives the bound $R_1+R_2\leq C_a+C_b+C_c.$