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

Xiangliang Kong, Shreya Meel, Thomas Jacob Maranzatto, Itzhak Tamo, Sennur Ulukus

TL;DR

The paper addresses private information retrieval (PIR) in graph- and multigraph-based replicated storage with two-copy files and pairwise overlap constraints, deriving sharp capacity bounds and constructing near-optimal schemes. Its key contributions are the exact PIR capacity for path graphs, improved bounds for complete bipartite graphs and complete graphs, and a symmetry-based multigraph construction that lifts graph schemes to $r$-multigraphs, yielding substantive lower bounds and several tight upper bounds. It also develops general reductions via graph decomposition to combine component schemes and extends capacity concepts to $r$-multigraphs, including asymptotic behavior as $r$ grows. The results advance understanding of PIR performance in sparse, structured storage networks and provide practical, symmetry-aware schemes for realistic distributed databases.

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.

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

TL;DR

The paper addresses private information retrieval (PIR) in graph- and multigraph-based replicated storage with two-copy files and pairwise overlap constraints, deriving sharp capacity bounds and constructing near-optimal schemes. Its key contributions are the exact PIR capacity for path graphs, improved bounds for complete bipartite graphs and complete graphs, and a symmetry-based multigraph construction that lifts graph schemes to -multigraphs, yielding substantive lower bounds and several tight upper bounds. It also develops general reductions via graph decomposition to combine component schemes and extends capacity concepts to -multigraphs, including asymptotic behavior as grows. The results advance understanding of PIR performance in sparse, structured storage networks and provide practical, symmetry-aware schemes for realistic distributed databases.

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 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.
Paper Structure (16 sections, 15 theorems, 104 equations, 3 figures, 5 tables)

This paper contains 16 sections, 15 theorems, 104 equations, 3 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Problem Setting
  4. PIR Reductions by Graph Decomposition
  5. Improved Capacity Bounds and Schemes for PIR over Graphs
  6. The PIR Capacity of Path Graphs
  7. Improved Capacity Bounds for Complete Bipartite Graphs
  8. Improved PIR Scheme for Complete Graphs
  9. Capacity Bounds and Scheme Constructions for PIR over Multigraphs
  10. A General PIR Scheme for $r$-Multigraphs
  11. Other Capacity Lower Bounds for $r$-Multigraphs
  12. Capacity Upper Bounds for PIR over $r$- Multigraphs
  13. Conclusion and Future Directions
  14. Proofs of Theorem \ref{['thm-stam']}
  15. Proof of Lemma \ref{["Variant of Lem11 in Zachi's paper"]}
  16. ...and 1 more sections

Key Result

Proposition 2.1

SGT23 Given a set of file indices $J \subseteq [K]$, we denote $W_{J} \triangleq \{W_{i} : i \in J\}$. Then, for any answer $A_i$, $i \in [N]$, any requested file index $k \in [K]$, and any $J \subseteq [K]$,

Figures (3)

  • Figure 1: The replication system based on $\mathbf{P}_3$
  • Figure 2: The replication system based on $\mathbf{K}_3$
  • Figure 3: The replication system based on $\mathbf{P}_3^{(2)}$

Theorems & Definitions (26)

  • Proposition 2.1
  • Theorem 2.2
  • Theorem 3.1
  • Lemma 3.2
  • Theorem 3.3
  • Example 1
  • Theorem 3.4
  • Remark 3.5
  • Lemma 3.6
  • Example 2
  • ...and 16 more