New Capacity Bounds for PIR on Graph and Multigraph-Based Replicated Storage

2025-04-29

AI Summary

This paper investigates the private information retrieval (PIR) capacity problem under structured replication across servers, where each file is stored on exactly two servers and any pair of servers jointly stores at most $r$ files—a constraint naturally modeled via graphs and multigraphs. Leveraging combinatorial design, graph-theoretic modeling, and information-theoretic analysis, we propose a symmetry-driven query construction framework. Our contributions include: (i) determining the exact PIR capacity for path graphs; (ii) tightening the known upper and lower bounds for complete bipartite and complete graphs; and (iii) establishing novel tight lower bounds and several general tight upper and lower bounds for multigraph-based replication topologies—achieving optimality in multiple cases. The proposed schemes asymptotically approach the theoretical capacity limits, thereby substantially expanding both the fundamental understanding and constructive paradigms of structured PIR.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Machine Learning: Structured Learning
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Security and Privacy: Large-scale security measurements
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics

Abstract

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 most $r$ files. 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.

Problem

• Derive PIR capacity bounds for graph-based storage systems
• Improve PIR capacity results for specific graph types
• Establish general bounds for multigraph-based PIR systems

Innovation

• Derives upper bounds on PIR capacity
• Constructs PIR schemes approaching bounds
• Leverages symmetry for multigraph schemes