Symmetry, Outer Bounds, and Code Constructions: A Computer-Aided Investigation on the Fundamental Limits of Caching

We present our recent effort, in the context of the caching systems, in developing a computer-aided approach in the investigation of information systems. Yeung's linear programming (LP) outer bound of the entropy space is our starting point, however our effort goes significantly beyond using it to prove information inequalities. A symmetry-reduced linear program is used to identify the boundary of the memory-transmission-rate tradeoff for several simple cases, for which we can obtain a set of tight outer bounds. General hypotheses on the optimal tradeoff are formed from these computed data, which are then analytically proved. This leads to a complete characterization of the optimal tradeoff for caching systems with only two users, and a partial characterization for systems with only two files. Next, we show that by carefully analyzing the joint entropy structure of the outer bounds for certain cases, a novel code construction can be reverse-engineered, and unachievability of linear codes can be proved for some other cases. Finally, we show that strong outer bounds can be computed through strategically relaxing the LP, which allows us to compute outer bounds for larger problem cases, despite the seemingly impossible computation scale.