In this paper, we study the problem of private information retrieval (PIR) in both graph-based and multigraph-based replication systems, where each file is stored on exactly two servers, and any pair of servers shares at mostrfiles. We derive upper bounds on the PIR capacity for such systems and construct PIR schemes that approach these bounds. For graph-based systems, we determine the exact PIR capacity for path graphs and improve upon existing results for complete bipartite graphs and complete graphs. For multigraph-based systems, we propose a PIR scheme that leverages the symmetry of the underlying graph-based construction, yielding a capacity lower bound for such multigraphs. Furthermore, we establish several general upper and lower bounds on the PIR capacity of multigraphs, which are tight in certain cases.

We introduce the problem of symmetric private information retrieval (SPIR) on replicated databases modeled by a simple graph. In this model, each vertex corresponds to a server, and a message is replicated on two servers if and only if there is an edge between them. We consider the setting where the server-side common randomness necessary to accomplish SPIR is also replicated at the servers according to the graph, and we call this as message-specific common randomness. In this setting, we establish a lower bound on the SPIR capacity, i.e., the maximum download rate, for general graphs, by proposing an achievable SPIR scheme. Next, we prove that, for any SPIR scheme to be feasible, the minimum size of message-specific randomness should be equal to the size of a message. Finally, by providing matching upper bounds, we derive the exact SPIR capacity for the class of path and regular graphs.

In a classification task, counterfactual explanations provide the minimum change needed for an input to be classified into a favorable class. We consider the problem of privately retrieving the exact closest counterfactual from a database of accepted samples while enforcing that certain features of the input sample cannot be changed, i.e., they are immutable. An applicant (user) whose feature vector is rejected by a machine learning model wants to retrieve the sample closest to them in the database without altering a private subset of their features, which constitutes the immutable set. While doing this, the user should keep their feature vector, immutable set and the resulting counterfactual index information-theoretically private from the institution. We refer to this as immutable private counterfactual retrieval (I-PCR) problem which generalizes PCR to a more practical setting. In this paper, we propose two I-PCR schemes by leveraging techniques from private information retrieval (PIR) and characterize their communication costs. Further, we quantify the information that the user learns about the database and compare it for the proposed schemes.

Verifying user attributes to provide fine-grained access control to databases is fundamental to an attribute-based authentication system. In such systems, either a single (central) authority verifies all attributes, or multiple independent authorities verify individual attributes distributedly to allow a user to access records stored on the servers. While a central setup is more communication cost efficient, it causes privacy breach of all user attributes to a central authority. Recently, Jafarpisheh et al. studied an information theoretic formulation of the distributed multi-authority setup with$N$non-colluding authorities,$N$attributes and$K$possible values for each attribute, called an$(N, K)$distributed attribute-based private access control (DAPAC) system, where each server learns only one attribute value that it verifies, and remains oblivious to the remaining$N-1$attributes. We show that off-loading a subset of attributes to a central server for verification improves the achievable rate from$\frac{1}{2K}$in Jafarpisheh et al. to$\frac{1}{K+1}$in this paper, thus almost doubling the rate for relatively large$K$, while sacrificing the privacy of a few possibly non-sensitive attributes.