Novel Impossibility Results for Group-Testing
Abhishek Agarwal, Sidharth Jaggi, Arya Mazumdar

TL;DR
This paper establishes fundamental lower bounds for non-adaptive group testing in the linear sparsity regime, revealing limitations of non-adaptive strategies and highlighting an adaptivity gap, using novel information-theoretic inequalities.
Contribution
It introduces the first lower bounds that surpass classical counting bounds for non-adaptive group testing in the linear regime, demonstrating an adaptivity gap and providing a new framework for non-linear estimation problems.
Findings
Non-adaptive testing requires nearly n tests for δ ≥ 0.347.
Adaptive algorithms outperform non-adaptive ones in certain sparsity regimes.
The work introduces a new framework combining combinatorial and information-theoretic methods.
Abstract
In this work we prove non-trivial impossibility results for perhaps the simplest non-linear estimation problem, that of {\it Group Testing} (GT), via the recently developed Madiman-Tetali inequalities. Group Testing concerns itself with identifying a hidden set of defective items from a set of items via {disjunctive/pooled} measurements ("group tests"). We consider the linear sparsity regime, i.e. for any constant , a hitherto little-explored (though natural) regime. In a standard information-theoretic setting, where the tests are required to be non-adaptive and a small probability of reconstruction error is allowed, our lower bounds on are the {\it first} that improve over the classical counting lower bound, , where is the binary entropy function. As corollaries of our result, we show that (i) for $\delta \gtrsim…
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