Competitive Group Testing and Learning Hidden Vertex Covers With Minimum Adaptivity

Abstract

Suppose that we are given a set of n elements d of which have a property called defective. A group test can check for any subset, called a pool, whether it contains a defective. It is known that a nearly optimal number of O(d log(n/d)) pools in two stages (where tests within a stage are done in parallel) are sufficient, but then the searcher must know d in advance. Here we explore group testing strategies that use a nearly optimal number of pools and a few stages although d is not known beforehand. We prove a lower bound of Ω(log d/ log log d) stages and more general pools versus stages tradeoff. This is almost tight, since O(log d) stages are sufficient for a strategy with O(d log n) pools. As opposed to this negative result, we devise a randomized strategy using O(d log(n/d)) pools in three stages, with any desired success probability 1 − . With some additional measures even two stages are enough. Open questions concern the optimal constant factors and practical implications. A related problem motivated by biological network analysis is to learn hidden vertex covers of a small size k in unknown graphs by edge group tests. (Does a given subset of vertices contain an edge?) We give a one-stage strategy using O(k 3 log n) pools, with any parameterized algorithm for vertex cover enumeration as a decoder. During the course of this work we also provide a classification of types of randomized search strategies in general. 84-95. 291 Discrete Math. Algorithm. Appl. 2010.02:291-311. Downloaded from www.worldscientific.com by UNIVERSITY OF AUCKLAND LIBRARY -SERIALS UNIT on 03/11/15. For personal use only. 292 P. Damaschke & A. S. Muhammad choose arbitrary subsets Q ⊂ X called pools, and ask whether Q contains at least one defective. Nondefective elements are called negative. A positive pool is a pool containing some defective, thus responding Yes to a group test. A negative pool is a pool without defectives, thus responding No to a group test.Group testing has several applications, most notably in biological and chemical testing, but also in communication networks, information gathering, compression, streaming algorithms, etc., see for instance [9,10,15,[20][21][22] and further pointers therein.Throughout this paper, log means log 2 if no other base is mentioned. By the information-theoretic lower bound, at least log n d ≈ d log(n/d) pools are needed to find d defectives even if the number d is known in advance, and it is an easy exercise to devise an adaptive query strategy using O(d log(n/d)) pools. Here, a strategy is called adaptive if queries are asked sequentially, that is, every pool can be prepared based on the outcomes of all earlier queries. For many applications however, the time consumption of adaptive strategies is hardly acceptable, and strategies that work in a few stages are strongly preferred: The pools for every stage must be prepared in advance, depending on the outcomes of earlier stages, and then they are queried in parallel.Any one-stage strategy needs Ω(d 2 log n/ log d) pools, as a consequence of ...

Large-scale implementation of pooled RNA-extraction and RT-PCR for SARS-CoV-2 detection

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“…for an integer s. The following result [55,Theorem 4.3] shows that (5.6) satisfies certain success criteria.…”

“…Constructions based on (5.2) for a particular p( ) will struggle to distinguish between putative values k 1 and k 2 that are both far from . To overcome this, Damaschke and Muhammad [55,Section 4] proposed a geometric construction, dividing the range of tests into subintervals of exponentially increasing size and using a series of p( ) values tailored to each subinterval.…”

“…Hence, if t * is the largest index of a negative pool, we may imagine that q t * 1/2 and invert this to construct an estimate of k. More precisely, [55] propose an offset of this, taking…”

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