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Optimal group testing

Part of: Probability theory on algebraic and topological structures Theory of data Graph theory

Published online by Cambridge University Press:  28 January 2021

Amin Coja-Oghlan ,
Oliver Gebhard ,
Max Hahn-Klimroth  and
Philipp Loick
Amin Coja-Oghlan*
Affiliation:
Goethe University Frankfurt, Robert-Mayer-Strasse 6–10, 60325 Frankfurt, Germany
Oliver Gebhard
Affiliation:
Goethe University Frankfurt, Robert-Mayer-Strasse 6–10, 60325 Frankfurt, Germany
Max Hahn-Klimroth
Affiliation:
Goethe University Frankfurt, Robert-Mayer-Strasse 6–10, 60325 Frankfurt, Germany
Philipp Loick
Affiliation:
Goethe University Frankfurt, Robert-Mayer-Strasse 6–10, 60325 Frankfurt, Germany
*
*Corresponding author. Email: [email protected]
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Abstract

In the group testing problem the aim is to identify a small set of k ⁓ nθ infected individuals out of a population size n, 0 < θ < 1. We avail ourselves of a test procedure capable of testing groups of individuals, with the test returning a positive result if and only if at least one individual in the group is infected. The aim is to devise a test design with as few tests as possible so that the set of infected individuals can be identified correctly with high probability. We establish an explicit sharp information-theoretic/algorithmic phase transition minf for non-adaptive group testing, where all tests are conducted in parallel. Thus with more than minf tests the infected individuals can be identified in polynomial time with high probability, while learning the set of infected individuals is information-theoretically impossible with fewer tests. In addition, we develop an optimal adaptive scheme where the tests are conducted in two stages.

MSC classification

Primary: 05C80: Random graphs
Secondary: 60B20: Random matrices (probabilistic aspects; for algebraic aspects see ) 68P30: Coding and information theory (compaction, compression, models of communication, encoding schemes, etc.)
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 30 , Issue 6 , November 2021 , pp. 811 - 848
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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Footnotes

†

Supported by DFG CO 646/3 and Stiftung Polytechnische Gesellschaft. An extended abstract version of this work appeared in the proceedings of the COLT 2020 conference (Proc. Mach. Learning Res. 125 (2020) 1374–1388).

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