Centralized coded caching schemes: A hypergraph theoretical approach

The centralized coded caching scheme is a technique proposed by Maddah-Ali and Niesen as a solution to reduce the network burden in peak times in a wireless network system. Yan et al. reformulate the problem as designing a corresponding placement delivery array and propose two new schemes from this perspective. These schemes above significantly reduce the transmission rate R, compared with the uncoded caching scheme. However, to implement the new schemes, each file should be cut into F pieces, where F increases exponentially with the number of users K. Such constraint is obviously infeasible in the practical setting, especially when K is large. Thus it is desirable to design caching schemes with constant rate R (independent of K) as well as small F . In this paper we view the centralized coded caching problem in a hypergraph perspective and show that designing a feasible placement delivery array is equivalent to constructing a linear and (6, 3)-free 3-uniform 3-partite hypergraph. Several new results and constructions arise from our novel point of view. First, by using the famous (6, 3)-theorem in extremal combinatorics, we show that constant rate caching schemes with F growing linearly with K do not exist. Second, we present two infinite classes of centralized coded caching schemes, which include the schemes of Ali-Niesen and Yan et al. as special cases, respectively. Moreover, our constructions show that constant rate caching schemes with F growing sub-exponentially with K do exist.