The Capacity of 4-Star-Graph PIR

Introduced by Sadeh et al., the K-star-graph private information retrieval (PIR) problem, so-labeled because the storage graph is a star-graph with K leaf nodes, is comprised of K messages that are stored separately (one-each) at K dedicated servers, and a universal server that stores all K messages, for a total of K + 1 servers. While it is one of the simplest PIR settings to describe, the capacity C
K
of K-star-graph PIR is open for K ≥ 4. We study the critical K = 4 setting, for which prior work establishes the bounds 2/5 ≤ C
4
≤ 3/7. As our main contribution, we characterize the exact capacity of 4-star-graph PIR as C
4
= 5/12, thus improving upon both the prior lower-bound as well as the prior upper-bound. The main technical challenge resides in the new converse bound, whose non-trivial structure is deduced indirectly from the achievable schemes that emerge from the study of a finer tradeoff between the download costs from the dedicated servers versus the universal server. A sharp characterization of this tradeoff is also obtained for K = 4.