The Capacity of Private Information Retrieval Under Arbitrary Collusion Patterns

We study the private information retrieval (PIR) problem under arbitrary collusion pattern for replicated databases. We find its capacity, which is the same as the capacity of the original PIR problem with the number of databases $N$ replaced by a number $S^{\ast }$. The number $S^{\ast }$ is the optimal solution to a linear programming problem that is a function of the collusion pattern. Hence, the collusion pattern affects the capacity of the PIR problem only through the number $S^{\ast }$.