Coded caching aims to minimize the network’s peak-time communication load by leveraging the information pre-stored in the local caches at the users. The original setting by Maddah-Ali and Niesen, which considered single file retrieval, has been recently extended to general Scalar Linear Function Retrieval (SLFR) by Wan et al., who proposed a linear scheme that surprisingly achieves the same optimal load under the constraint of uncoded cache placement as in single file retrieval. This paper’s goal is to characterize the conditions under which a general SLFR linear scheme is optimal and gain insights into why the specific choices made by Wan et al. work. This paper shows that the optimal decoding coefficients are necessarily the product of two terms, one only involving the encoding coefficients and the other only the demands of the users. In addition, the algebraic relationships among the encoding coefficients of an optimal code are shown to be captured by the cycles of a universal graph. Thus, a general linear scheme for the SLFR problem can be found by solving a spanning tree problem for the universal graph. The proposed framework readily extends to caching-like problems, such as the problem of finding a general linear scheme for Sun et al.’s private function computation.
