Bounds on the Capacity of PIR over Graphs

TL;DR

This paper investigates the maximum achievable rate of private information retrieval in graph-based storage systems, providing bounds and schemes that depend on graph properties, with improvements for specific graph classes.

Contribution

It introduces new upper and lower bounds on PIR capacity for graph-based systems, including optimal schemes for star and complete graphs.

Findings

Derived upper bounds applicable to all graphs.

Established optimal PIR schemes for star graphs.

Improved PIR schemes for complete graphs.

Abstract

In the private information retrieval (PIR) problem, a user wants to retrieve a file from a database without revealing any information about the desired file's identity to the servers that store the database. In this paper, we study the PIR capacity of a graph-based replication system, in which each file is stored on two distinct servers according to an underlying graph. This paper aims to provide upper and lower bounds to the PIR capacity of graphs via various graph properties. In particular, we provide several upper bounds on the PIR capacity that apply to all graphs. We further improve the bounds for specific graph families (which turn out to be tight in certain cases) by utilizing the underlying graph structure. For the lower bounds, we establish optimal rate PIR retrieval schemes for star graphs via edge-coloring techniques. Lastly, we provide an improved PIR scheme for complete graphs, which implies an improved general lower bound on all graphs' PIR capacity.