On the Capacity of Computation Broadcast

The two-user computation broadcast problem is introduced as the setting where User $1$ wants message $W_1$ and has side-information $W_1'$, User $2$ wants message $W_2$ and has side-information $W_2'$, and $(W_1, W_1', W_2, W_2')$ may have arbitrary dependencies. The rate of a computation broadcast scheme is defined as the ratio $H(W_1,W_2)/H(S)$, where $S$ is the information broadcast to both users to simultaneously satisfy their demands. The supremum of achievable rates is called the capacity of computation broadcast $C_{{CB}}$. It is shown that $C_{{CB}}\leq H(W_1,W_2)/\left[H(W_1|W_1')+H(W_2|W_2')-\min\Big(I(W_1; W_2, W_2'|W_1'), I(W_2; W_1, W_1'|W_2')\Big)\right]$. For the linear computation broadcast problem, where $W_1, W_1', W_2, W_2'$ are comprised of arbitrary linear combinations of a basis set of independent symbols, the bound is shown to be tight. For non-linear computation broadcast, it is shown that this bound is not tight in general. Examples are provided to prove that different instances of computation broadcast that have the same entropic structure, i.e., the same entropy for all subsets of $\{W_1,W_1',W_2,W_2'\}$, can have different capacities. Thus, extra-entropic structure matters even for two-user computation broadcast. The significance of extra-entropic structure is further explored through a class of non-linear computation broadcast problems where the extremal values of capacity are shown to correspond to minimally and maximally structured problems within that class.