The Capacity of Private Information Retrieval Under Arbitrary Collusion Patterns for Replicated Databases

We study the private information retrieval (PIR) problem under arbitrary collusion patterns for replicated databases. We find a general characterization of the PIR capacity, which is the same as the capacity of the original PIR problem with the number of databases N replaced by a number S^*. S^* is the optimal solution to a linear programming problem that is a function of the incidence matrix of the collusion pattern. Hence, the essence of any collusion pattern can be distilled into one number S^*. In the proposed achievable scheme, databases are non-uniformly queried according to the optimal solution of a linear programming problem based on the collusion pattern. It can be seen that the databases who collude more with others are queried less. In the converse proof, Shearer's lemma is applied, in place of Han's inequality, to a linear combination of inequalities, where each inequality corresponds to one colluding set in the collusion pattern. The weights of the linear combination come from the optimal solution of another linear programming problem based on the collusion pattern. Finally, by noting the interesting fact that the two seemingly different linear programming problems, one used in the achievability proof and the other used in the converse proof, are in fact dual problems, we characterize the capacity of the PIR problem under arbitrary collusion patterns.