Lower bounds for identifying subset members with subset queries

An instance of a group testing problem is a set of objects \cO and an unknown subset $P$ of \cO. The task is to determine $P$ by using queries of the type ``does $P$ intersect $Q$'', where $Q$ is a subset of \cO. This problem occurs in areas such as fault detection, multiaccess communications, optimal search, blood testing and chromosome mapping. Consider the two stage algorithm for solving a group testing problem. In the first stage a predetermined set of queries are asked in parallel and in the second stage, $P$ is determined by testing individual objects. Let n=\cardof{\cO}. Suppose that $P$ is generated by independently adding each x\in \cO to $P$ with probability $p/n$. Let $q_1$ ($q_2$) be the number of queries asked in the first (second) stage of this algorithm. We show that if $q_1=o(\log(n)\log(n)/\log\log(n))$, then \Exp(q_2) = n^{1-o(1)}, while there exist algorithms with $q_1 = O(\log(n)\log(n)/\log\log(n))$ and \Exp(q_2) = o(1). The proof involves a relaxation technique which can be used with arbitrary distributions. The best previously known bound is q_1+\Exp(q_2) = \Omega(p\log(n)). For general group testing algorithms, our results imply that if the average number of queries over the course of $n^\gamma$ ($\gamma>0$) independent experiments is $O(n^{1-\epsilon})$, then with high probability $\Omega(\log(n)\log(n)/\log\log(n))$ non-singleton subsets are queried. This settles a conjecture of Bill Bruno and David Torney and has important consequences for the use of group testing in screening DNA libraries and other applications where it is more cost effective to use non-adaptive algorithms and/or too expensive to prepare a subset $Q$ for its first test.Comment: 9 page

Group testing problems in experimental molecular biology

In group testing, the task is to determine the distinguished members of a set of objects L by asking subset queries of the form ``does the subset Q of L contain a distinguished object?'' The primary biological application of group testing is for screening libraries of clones with hybridization probes. This is a crucial step in constructing physical maps and for finding genes. Group testing has also been considered for sequencing by hybridization. Another important application includes screening libraries of reagents for useful chemically active zones. This preliminary report discusses some of the constrained group testing problems which arise in biology.Comment: 7 page

Efficient Two-Stage Group Testing Algorithms for Genetic Screening

Efficient two-stage group testing algorithms that are particularly suited for rapid and less-expensive DNA library screening and other large scale biological group testing efforts are investigated in this paper. The main focus is on novel combinatorial constructions in order to minimize the number of individual tests at the second stage of a two-stage disjunctive testing procedure. Building on recent work by Levenshtein (2003) and Tonchev (2008), several new infinite classes of such combinatorial designs are presented.Comment: 14 pages; to appear in "Algorithmica". Part of this work has been presented at the ICALP 2011 Group Testing Workshop; arXiv:1106.368

Derandomization and Group Testing

The rapid development of derandomization theory, which is a fundamental area in theoretical computer science, has recently led to many surprising applications outside its initial intention. We will review some recent such developments related to combinatorial group testing. In its most basic setting, the aim of group testing is to identify a set of "positive" individuals in a population of items by taking groups of items and asking whether there is a positive in each group. In particular, we will discuss explicit constructions of optimal or nearly-optimal group testing schemes using "randomness-conducting" functions. Among such developments are constructions of error-correcting group testing schemes using randomness extractors and condensers, as well as threshold group testing schemes from lossless condensers.Comment: Invited Paper in Proceedings of 48th Annual Allerton Conference on Communication, Control, and Computing, 201

Explicit Non-Adaptive Combinatorial Group Testing Schemes

Group testing is a long studied problem in combinatorics: A small set of $r$ ill people should be identified out of the whole ($n$ people) by using only queries (tests) of the form "Does set X contain an ill human?". In this paper we provide an explicit construction of a testing scheme which is better (smaller) than any known explicit construction. This scheme has \bigT{\min[r^2 \ln n,n]} tests which is as many as the best non-explicit schemes have. In our construction we use a fact that may have a value by its own right: Linear error-correction codes with parameters $[m,k,\delta m]_q$ meeting the Gilbert-Varshamov bound may be constructed quite efficiently, in \bigT{q^km} time.Comment: 15 pages, accepted to ICALP 200

Noise-Resilient Group Testing: Limitations and Constructions

We study combinatorial group testing schemes for learning $d$-sparse Boolean vectors using highly unreliable disjunctive measurements. We consider an adversarial noise model that only limits the number of false observations, and show that any noise-resilient scheme in this model can only approximately reconstruct the sparse vector. On the positive side, we take this barrier to our advantage and show that approximate reconstruction (within a satisfactory degree of approximation) allows us to break the information theoretic lower bound of $\tilde{\Omega}(d^2 \log n)$ that is known for exact reconstruction of $d$-sparse vectors of length $n$ via non-adaptive measurements, by a multiplicative factor $\tilde{\Omega}(d)$. Specifically, we give simple randomized constructions of non-adaptive measurement schemes, with $m=O(d \log n)$ measurements, that allow efficient reconstruction of $d$-sparse vectors up to $O(d)$ false positives even in the presence of $\delta m$ false positives and $O(m/d)$ false negatives within the measurement outcomes, for any constant $\delta < 1$. We show that, information theoretically, none of these parameters can be substantially improved without dramatically affecting the others. Furthermore, we obtain several explicit constructions, in particular one matching the randomized trade-off but using $m = O(d^{1+o(1)} \log n)$ measurements. We also obtain explicit constructions that allow fast reconstruction in time \poly(m), which would be sublinear in $n$ for sufficiently sparse vectors. The main tool used in our construction is the list-decoding view of randomness condensers and extractors.Comment: Full version. A preliminary summary of this work appears (under the same title) in proceedings of the 17th International Symposium on Fundamentals of Computation Theory (FCT 2009

Application of cover-free codes and combinatorial designs to two-stage testing

AbstractWe study combinatorial and probabilistic properties of cover-free codes and block designs which are useful for their efficient application as the first stage of two-stage group testing procedures. Particular attention is paid to these procedures because of their importance in such applications as monoclonal antibody generation and cDNA library screening

Constraining the Number of Positive Responses in Adaptive, Non-Adaptive, and Two-Stage Group Testing

Group testing is a well known search problem that consists in detecting the defective members of a set of objects O by performing tests on properly chosen subsets (pools) of the given set O. In classical group testing the goal is to find all defectives by using as few tests as possible. We consider a variant of classical group testing in which one is concerned not only with minimizing the total number of tests but aims also at reducing the number of tests involving defective elements. The rationale behind this search model is that in many practical applications the devices used for the tests are subject to deterioration due to exposure to or interaction with the defective elements. In this paper we consider adaptive, non-adaptive and two-stage group testing. For all three considered scenarios, we derive upper and lower bounds on the number of "yes" responses that must be admitted by any strategy performing at most a certain number t of tests. In particular, for the adaptive case we provide an algorithm that uses a number of "yes" responses that exceeds the given lower bound by a small constant. Interestingly, this bound can be asymptotically attained also by our two-stage algorithm, which is a phenomenon analogous to the one occurring in classical group testing. For the non-adaptive scenario we give almost matching upper and lower bounds on the number of "yes" responses. In particular, we give two constructions both achieving the same asymptotic bound. An interesting feature of one of these constructions is that it is an explicit construction. The bounds for the non-adaptive and the two-stage cases follow from the bounds on the optimal sizes of new variants of d-cover free families and (p,d)-cover free families introduced in this paper, which we believe may be of interest also in other contexts