Journal of Combinatorial Optimization

, Volume 35, Issue 4, pp 1042–1060 | Cite as

Smart elements in combinatorial group testing problems

Article
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Abstract

In combinatorial group testing problems the questioner needs to find a special element \(x \in [n]\) by testing subsets of [n]. Tapolcai et al. (in: Proceedings of IEEE INFOCOM, Toronto, Canada, pp 1860–1868, 2014; IEEE Trans Commun 64(6):2527–2538, 2016) introduced a new model, where each element knows the answer for those queries that contain it and each element should be able to identify the special one. Using classical results of extremal set theory we prove that if \(\mathcal {F}_n \subseteq 2^{[n]}\) solves the non-adaptive version of this problem and has minimal cardinality, then
$$\begin{aligned} \lim _{n \rightarrow \infty } \frac{|\mathcal {F}_n|}{\log _2 n} = \log _{(3/2)}2. \end{aligned}$$
This improves results in Tapolcai et al. (2014, 2016). We also consider related models inspired by secret sharing models, where the elements should share information among them to find out the special one. Finally the adaptive versions of the different models are investigated.

Keywords

Combinatorial group testing Non-adaptive Information sharing 

Notes

Acknowledgements

We would like to thank Éva Hosszu (2015), who asked us the first question of the type that was investigated in this article. We would also like to thank all participants of the Combinatorial Search Seminar at the Alfréd Rényi Institute of Mathematics for fruitful discussions. We also thank the anonymous reviewers for their careful reading of our manuscript and their many insightful comments and suggestions that improved the presentation of our article.

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.MTA Rényi InstituteBudapestHungary

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