The Capacity of Bernoulli Nonadaptive Group Testing

We consider nonadaptive group testing with Bernoulli tests, where each item is placed in each test independently with some fixed probability. We give a tight threshold on the maximum number of tests required to find the defective set under optimal Bernoulli testing. Achievability is given by a result of Scarlett and Cevher; here we give a converse bound showing that this result is best possible. Our new converse requires three parts: a typicality bound generalising the trivial counting bound, a converse on the COMP algorithm of Chan et al., and a bound on the SSS algorithm similar to that given by Aldridge, Baldassini, and Johnson. Our result has a number of important corollaries, in particular that, in denser cases, Bernoulli nonadaptive group testing is strictly worse than the best adaptive strategies.

[1]  J. Scarlett,et al.  Performance of group testing algorithms with constant tests-per-item , 2016 .

[2]  Matthew Aldridge,et al.  Performance of Group Testing Algorithms With Near-Constant Tests Per Item , 2016, IEEE Transactions on Information Theory.

[3]  A. Sebő.  ON TWO RANDOM SEARCH PROBLEMS , 1985 .

[4]  M. Mézard,et al.  Group Testing with Random Pools: Phase Transitions and Optimal Strategy , 2007, 0711.2242.

[5]  Matthew Aldridge,et al.  Group Testing Algorithms: Bounds and Simulations , 2013, IEEE Transactions on Information Theory.

[6]  D. Du,et al.  Combinatorial Group Testing and Its Applications , 1993 .

[7]  George Atia,et al.  Boolean Compressed Sensing and Noisy Group Testing , 2009, IEEE Transactions on Information Theory.

[8]  Venkatesh Saligrama,et al.  Non-adaptive probabilistic group testing with noisy measurements: Near-optimal bounds with efficient algorithms , 2011, 2011 49th Annual Allerton Conference on Communication, Control, and Computing (Allerton).

[9]  F. Hwang.  A Method for Detecting all Defective Members in a Population by Group Testing , 1972 .

[10]  Thomas M. Cover,et al.  Elements of Information Theory: Cover/Elements of Information Theory, Second Edition , 2005 .

[11]  Mikhail B. Malyutov,et al.  Search for Sparse Active Inputs: A Review , 2013, Information Theory, Combinatorics, and Search Theory.

[12]  Venkatesh Saligrama,et al.  Non-adaptive group testing: Explicit bounds and novel algorithms , 2012, 2012 IEEE International Symposium on Information Theory Proceedings.

[13]  Matthew Aldridge,et al.  The capacity of adaptive group testing , 2013, 2013 IEEE International Symposium on Information Theory.

[14]  Richard C. Singleton,et al.  Nonrandom binary superimposed codes , 1964, IEEE Trans. Inf. Theory.

[15]  Frank K. Hwang,et al.  Exploring the missing link among d-separable, d_-separable and d-disjunct matrices , 2007, Discret. Appl. Math..

[16]  Matthew Aldridge,et al.  Almost separable matrices , 2017, J. Comb. Optim..

[17]  Volkan Cevher,et al.  Limits on Support Recovery With Probabilistic Models: An Information-Theoretic Framework , 2015, IEEE Transactions on Information Theory.

[18]  M. Malyutov.  The separating property of random matrices , 1978 .

[19]  Matthew Aldridge.  On the optimality of some group testing algorithms , 2017, 2017 IEEE International Symposium on Information Theory (ISIT).

[20]  Frank K. Hwang,et al.  A survey on nonadaptive group testing algorithms through the angle of decoding , 2008, J. Comb. Optim..

[21]  Volkan Cevher,et al.  Phase Transitions in Group Testing , 2016, SODA.