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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 CK 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 ≤ C4 ≤ 3/7. As our main contribution, we characterize the exact capacity of 4-star-graph PIR as C4 = 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. 

Motivated by an open problem and a conjecture, this work studies the problem of single server private information retrieval with private coded side information (PIR-PCSI) that was recently introduced by Heidarzadeh et al. The goal of PIR-PCSI is to allow a user to efficiently retrieve a desired message Wθ, which is one of K independent messages that are stored at a server, while utilizing private side information of a linear combination of a uniformly chosen size-M subset (S ⊂ [K]) of messages. The settings PIR-PCSI-I and PIR-PCSI-II correspond to the constraints that θ is generated uniformly from [K]\S, and S, respectively. In each case, (θ, S) must be kept private from the server. The capacity is defined as the supremum over message and field sizes, of achievable rates (number of bits of desired message retrieved per bit of download) and is characterized by Heidarzadeh et al. for PIR-PCSI-I in general, and for PIR- PCSI-II for M > (K + 1)/2 as (K − M + 1)−1. For 2 ≤ M ≤ (K + 1)/2 the capacity of PIR-PCSI-II remains open, and it is conjectured that even in this case the capacity is (K − M + 1)−1. We show the capacity of PIR-PCSI-II is equal to 2/K for 2 ≤ M ≤ K+1, which is strictly larger 2 than the conjectured value, and does not depend on M within this parameter regime. Remarkably, half the side-information is found to be redundant. We also characterize the infimum capacity (infimum over fields instead of supremum), and the capacity with private coefficients. The results are generalized to PIR-PCSI-I (θ ∈ [K] \ S) and PIR-PCSI (θ ∈ [K]) settings.