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On Subset Retrieval and Group Testing Problems with Differential Privacy Constraints

Mira Gonen, Michael Langberg, Alex Sprintson

TL;DR

The paper develops differential privacy frameworks for two infection-status problems: DPSR and group testing. It defines DP guarantees via a detailed $(\varepsilon_1,\varepsilon_2,\delta,\beta)_{n,d}$ specification and characterizes the fundamental privacy-accuracy trade-offs, providing tight upper and lower bounds on $\delta$ and concrete schemes (DPSR via randomified set retrieval and DPSR-I via e-augmented randomization). It then extends these results to differential private group testing, showing that the subset-retrieval insights carry over to pool-based testing with two noise models, and provides practical implementation guidance for a privacy-preserving testing workflow. The work yields both theoretical bounds and constructive algorithms, with implications for privacy-preserving disease surveillance and participation-supported testing programs. It also outlines future directions, including hypergraph-restricted contaminated subsets and differential-private property testing and community detection.

Abstract

This paper focuses on the design and analysis of privacy-preserving techniques for group testing and infection status retrieval. Our work is motivated by the need to provide accurate information on the status of disease spread among a group of individuals while protecting the privacy of the infection status of any single individual involved. The paper is motivated by practical scenarios, such as controlling the spread of infectious diseases, where individuals might be reluctant to participate in testing if their outcomes are not kept confidential. The paper makes the following contributions. First, we present a differential privacy framework for the subset retrieval problem, which focuses on sharing the infection status of individuals with administrators and decision-makers. We characterize the trade-off between the accuracy of subset retrieval and the degree of privacy guaranteed to the individuals. In particular, we establish tight lower and upper bounds on the achievable level of accuracy subject to the differential privacy constraints. We then formulate the differential privacy framework for the noisy group testing problem in which noise is added either before or after the pooling process. We establish a reduction between the private subset retrieval and noisy group testing problems and show that the converse and achievability schemes for subset retrieval carry over to differentially private group testing.

On Subset Retrieval and Group Testing Problems with Differential Privacy Constraints

TL;DR

The paper develops differential privacy frameworks for two infection-status problems: DPSR and group testing. It defines DP guarantees via a detailed specification and characterizes the fundamental privacy-accuracy trade-offs, providing tight upper and lower bounds on and concrete schemes (DPSR via randomified set retrieval and DPSR-I via e-augmented randomization). It then extends these results to differential private group testing, showing that the subset-retrieval insights carry over to pool-based testing with two noise models, and provides practical implementation guidance for a privacy-preserving testing workflow. The work yields both theoretical bounds and constructive algorithms, with implications for privacy-preserving disease surveillance and participation-supported testing programs. It also outlines future directions, including hypergraph-restricted contaminated subsets and differential-private property testing and community detection.

Abstract

This paper focuses on the design and analysis of privacy-preserving techniques for group testing and infection status retrieval. Our work is motivated by the need to provide accurate information on the status of disease spread among a group of individuals while protecting the privacy of the infection status of any single individual involved. The paper is motivated by practical scenarios, such as controlling the spread of infectious diseases, where individuals might be reluctant to participate in testing if their outcomes are not kept confidential. The paper makes the following contributions. First, we present a differential privacy framework for the subset retrieval problem, which focuses on sharing the infection status of individuals with administrators and decision-makers. We characterize the trade-off between the accuracy of subset retrieval and the degree of privacy guaranteed to the individuals. In particular, we establish tight lower and upper bounds on the achievable level of accuracy subject to the differential privacy constraints. We then formulate the differential privacy framework for the noisy group testing problem in which noise is added either before or after the pooling process. We establish a reduction between the private subset retrieval and noisy group testing problems and show that the converse and achievability schemes for subset retrieval carry over to differentially private group testing.
Paper Structure (12 sections, 10 theorems, 66 equations)

This paper contains 12 sections, 10 theorems, 66 equations.

Table of Contents

  1. Introduction
  2. Model and Statement of Results
  3. Proof of Theorem \ref{['the:main-1']}
  4. Proof of Theorem \ref{['the:main-1']}: Upper bound on $\delta$
  5. Proof of Theorem \ref{['the:main-1']}: Lower bound on $\delta$
  6. Proof of Theorem \ref{['the:main2']}
  7. Theorem \ref{['the:main2']}: Upper bound on $\delta$
  8. Theorem \ref{['the:main2']}: Lower bound on $\delta$
  9. Group Testing with Differential Privacy
  10. Group Testing Model
  11. Implementation at the specimen collector
  12. Conclusion and Future Research

Key Result

Theorem 1

Any algorithm ${\cal A}$ that is $(\varepsilon_1,\varepsilon_2,\delta,\beta)_{n,d}$ differentially-private for $\beta^2 \leq n$ must satisfy Furthermore, there exists an $(\varepsilon_1,\varepsilon_2,\delta,\beta)_{n,d}$ differentially-private algorithm for $\varepsilon_1=0$, $\varepsilon_2=0$, and

Theorems & Definitions (12)

  • Definition 1
  • Theorem 1
  • Theorem 2
  • Lemma 1: Upper bound of Theorem \ref{['the:main-1']}
  • Lemma 2
  • Lemma 3
  • Lemma 4: Lower bound of Theorem \ref{['the:main-1']}
  • Lemma 5: Upper bound of Theorem \ref{['the:main2']}
  • Lemma 6
  • Lemma 7: Lower bound of Theorem \ref{['the:main2']}
  • ...and 2 more