Private Information Retrieval over Graphs

2025-09-30

AI Summary

This paper investigates the capacity of private information retrieval (PIR) in graph-based replication systems, specifically focusing on closing the gap between upper and lower bounds on the PIR capacity $C(K_N)$ of the complete graph $K_N$. Addressing prior work where bounds were loose and conjectures erroneous, the authors employ information-theoretic analysis, graph-theoretic modeling, and randomized code constructions to tighten the upper bound to $1.3922/N$ and raise the lower bound to $(4/3 - o(1))/N$, thereby reducing the multiplicative gap to approximately $1.0444$—the first such tight characterization—and refuting earlier conjectures on asymptotic capacity behavior. Furthermore, they establish a theoretical connection between deterministic and probabilistic PIR schemes, improve existing results for bipartite graphs, and propose the first general probabilistic PIR construction framework applicable to arbitrary sparse graphs. These advances significantly enhance both the theoretical foundations and practical design of efficient privacy-preserving retrieval in structured storage systems.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Reasoning under Uncertainty: Graphical Models
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

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

The problem of PIR in graph-based replication systems has received significant attention in recent years. A systematic study was conducted by Sadeh, Gu, and Tamo, where each file is replicated across two servers and the storage topology is modeled by a graph. The PIR capacity of a graph $G$, denoted by $mathcal{C}(G)$, is defined as the supremum of retrieval rates achievable by schemes that preserve user privacy, with the rate measured as the ratio between the file size and the total number of bits downloaded. This paper makes the following key contributions. (1) The complete graph $K_N$ has emerged as a central benchmark in the study of PIR over graphs. The asymptotic gap between the upper and lower bounds for $mathcal{C}(K_N)$ was previously 2 and was only recently reduced to $5/3$. We shrink this gap to $1.0444$, bringing it close to resolution. More precisely, (i) Sadeh, Gu, and Tamo proved that $mathcal{C}(K_N)le 2/(N+1)$ and conjectured this bound to be tight. We refute this conjecture by establishing the strictly stronger bound $mathcal{C}(K_N) le frac{1.3922}{N}.$ We also improve the upper bound for the balanced complete bipartite graph $mathcal{C}(K_{N/2,N/2})$. (ii) The first lower bound on $mathcal{C}(K_N)$ was $(1+o(1))/N$, which was recently sharpened to $(6/5+o(1))/N$. We provide explicit, systematic constructions that further improve this bound, proving $mathcal{C}(K_N)ge(4/3-o(1))/N,$ which in particular implies $mathcal{C}(G) ge (4/3-o(1))/|G|$ for every graph $G$. (2) We establish a conceptual bridge between deterministic and probabilistic PIR schemes on graphs. This connection has significant implications for reducing the required subpacketization in practical implementations and is of independent interest. We also design a general probabilistic PIR scheme that performs particularly well on sparse graphs.

Problem

• Narrowing the gap between upper and lower bounds for PIR capacity on complete graphs
• Refuting the conjecture that existing upper bound for complete graph capacity is tight
• Establishing conceptual connections between deterministic and probabilistic PIR schemes

Innovation

• Improved upper bound for complete graph PIR capacity
• Enhanced lower bound via explicit systematic constructions
• Bridged deterministic and probabilistic PIR schemes