The $K$ User Linear Computation Broadcast (LCBC) problem is comprised of $d$ dimensional data (from $\mathbb{F}_q$), that is fully available to a central server, and $K$ users, who require various linear computations of the data, and have prior knowledge of various linear functions of the data as side-information. The optimal broadcast cost is the minimum number of $q$-ary symbols to be broadcast by the server per computation instance, for every user to retrieve its desired computation. The reciprocal of the optimal broadcast cost is called the capacity. The main contribution of this paper is the exact capacity characterization for the $K=3$ user LCBC for all cases, i.e., for arbitrary finite fields $\mathbb{F}_q$, arbitrary data dimension $d$, and arbitrary linear side-informations and demands at each user. A remarkable aspect of the converse is that unlike the $2$ user LCBC whose capacity was determined previously, the entropic formulation (where the entropies of demands and side-informations are specified, but not their functional forms) is insufficient to obtain a tight converse for the $3$ user LCBC. Instead, the converse exploits functional submodularity. Notable aspects of achievability include a decomposition of the users' collective signal space into subspaces that allow different degrees of efficiency in broadcast cost, revealing a tradeoff that leads to a constrained water-filling solution. Random coding arguments are invoked to resolve compatibility issues that arise as each user has a different view of these subspaces, conditioned on its own side-information.

翻译：K用户线性计算广播（LCBC）问题包含d维数据（来自有限域$\mathbb{F}_q$），该数据完全由中央服务器掌握，K个用户需要计算数据的各种线性函数，并预先知道数据的若干线性函数作为边信息。最优广播成本是指每个用户为获取其所需计算任务时，服务器每次计算实例需广播的最小q进制符号数。最优广播成本的倒数称为容量。本文的主要贡献在于对所有情况下的K=3用户LCBC问题给出了精确容量刻画，即针对任意有限域$\mathbb{F}_q$、任意数据维度d以及各用户任意线性边信息和需求。反方向证明的一个显著特点是：与先前确定容量的2用户LCBC不同，熵公式化方法（仅指定需求和边信息的熵值，而非其函数形式）不足以获得3用户LCBC的紧致反方向证明。相反，该反方向证明利用了函数子模性。可达性方面的显著特征包括：将用户的联合信号空间分解为允许不同广播成本效率的子空间，揭示了导致约束注水解的权衡。为解决各用户基于自身边信息对这些子空间具有不同视角而产生的兼容性问题，本文采用了随机编码论证。