Some new bounds for cover-free families
Journal of Combinatorial Theory Series A
Randomness conductors and constant-degree lossless expanders
STOC '02 Proceedings of the thiry-fourth annual ACM symposium on Theory of computing
Unbalanced Expanders and Randomness Extractors from Parvaresh-Vardy Codes
CCC '07 Proceedings of the Twenty-Second Annual IEEE Conference on Computational Complexity
Explicit Non-adaptive Combinatorial Group Testing Schemes
ICALP '08 Proceedings of the 35th international colloquium on Automata, Languages and Programming, Part I
Nonadaptive algorithms for threshold group testing
Discrete Applied Mathematics
Noise-resilient group testing: limitations and constructions
FCT'09 Proceedings of the 17th international conference on Fundamentals of computation theory
Reconstruction of hidden graphs and threshold group testing
Journal of Combinatorial Optimization
General Theory of Information Transfer and Combinatorics
Threshold and majority group testing
Information Theory, Combinatorics, and Search Theory
Superimposed codes and threshold group testing
Information Theory, Combinatorics, and Search Theory
Group testing with multiple mutually-obscuring positives
Information Theory, Combinatorics, and Search Theory
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The basic goal in combinatorial group testing is to identify a set of up to d defective items within a large population of size n ≫ d using a pooling strategy. Namely, the items can be grouped together in pools, and a single measurement would reveal whether there are one or more defectives in the pool. The threshold model is a generalization of this idea where a measurement returns positive if the number of defectives in the pool passes a fixed threshold u, negative if this number is below a fixed lower threshold l ≤ u, and may behave arbitrarily otherwise. We study non-adaptive threshold group testing (in a possibly noisy setting) and show that, for this problem, O(dg+2(log d) log(n/d)) measurements (where g := u - l) suffice to identify the defectives, and also present almost matching lower bounds. This significantly improves the previously known (non-constructive) upper bound O(du+1log(n/d)). Moreover, we obtain a framework for explicit construction of measurement schemes using lossless condensers. The number of measurements resulting from this scheme is ideally bounded by O(dg+3(log d) log n). Using state-of-the-art constructions of lossless condensers, however, we come up with explicit testing schemes with O(dg+3(log d)quasipoly(log n)) and O(dg+3+βpoly(log n)) measurements, for arbitrary constant β