Concomitant Group Testing

In this paper, we introduce a variation of the group testing problem capturing the idea that a positive test requires a combination of multiple ``types'' of item. Specifically, we assume that there are multiple disjoint \emph{semi-defective sets}, and a test is positive if and only if it contains at least one item from each of these sets. The goal is to reliably identify all of the semi-defective sets using as few tests as possible, and we refer to this problem as \textit{Concomitant Group Testing} (ConcGT). We derive a variety of algorithms for this task, focusing primarily on the case that there are two semi-defective sets. Our algorithms are distinguished by (i) whether they are deterministic (zero-error) or randomized (small-error), and (ii) whether they are non-adaptive, fully adaptive, or have limited adaptivity (e.g., 2 or 3 stages). Both our deterministic adaptive algorithm and our randomized algorithms (non-adaptive or limited adaptivity) are order-optimal in broad scaling regimes of interest, and improve significantly over baseline results that are based on solving a more general problem as an intermediate step (e.g., hypergraph learning).Comment: 15 pages, 3 figures, 1 tabl

Nearly Optimal Sparse Group Testing

Group testing is the process of pooling arbitrary subsets from a set of $n$ items so as to identify, with a minimal number of tests, a "small" subset of $d$ defective items. In "classical" non-adaptive group testing, it is known that when $d$ is substantially smaller than $n$, $\Theta(d\log(n))$ tests are both information-theoretically necessary and sufficient to guarantee recovery with high probability. Group testing schemes in the literature meeting this bound require most items to be tested $\Omega(\log(n))$ times, and most tests to incorporate $\Omega(n/d)$ items. Motivated by physical considerations, we study group testing models in which the testing procedure is constrained to be "sparse". Specifically, we consider (separately) scenarios in which (a) items are finitely divisible and hence may participate in at most $\gamma \in o(\log(n))$ tests; or (b) tests are size-constrained to pool no more than $\rho \in o(n/d)$items per test. For both scenarios we provide information-theoretic lower bounds on the number of tests required to guarantee high probability recovery. In both scenarios we provide both randomized constructions (under both $\epsilon$-error and zero-error reconstruction guarantees) and explicit constructions of designs with computationally efficient reconstruction algorithms that require a number of tests that are optimal up to constant or small polynomial factors in some regimes of $n, d, \gamma,$ and $\rho$. The randomized design/reconstruction algorithm in the $\rho$-sized test scenario is universal -- independent of the value of $d$, as long as $\rho \in o(n/d)$. We also investigate the effect of unreliability/noise in test outcomes. For the full abstract, please see the full text PDF

On Detecting Some Defective Items in Group Testing

Group testing is an approach aimed at identifying up to $d$ defective items among a total of $n$ elements. This is accomplished by examining subsets to determine if at least one defective item is present. In our study, we focus on the problem of identifying a subset of $\ell\leq d$ defective items. We develop upper and lower bounds on the number of tests required to detect $\ell$ defective items in both the adaptive and non-adaptive settings while considering scenarios where no prior knowledge of $d$ is available, and situations where an estimate of $d$ or at least some non-trivial upper bound on $d$ is available. When no prior knowledge on $d$ is available, we prove a lower bound of $\Omega(\frac{\ell \log^2n}{\log \ell +\log\log n})$ tests in the randomized non-adaptive settings and an upper bound of $O(\ell \log^2 n)$ for the same settings. Furthermore, we demonstrate that any non-adaptive deterministic algorithm must ask $\Theta(n)$ tests, signifying a fundamental limitation in this scenario. For adaptive algorithms, we establish tight bounds in different scenarios. In the deterministic case, we prove a tight bound of $\Theta(\ell\log{(n/\ell)})$. Moreover, in the randomized settings, we derive a tight bound of $\Theta(\ell\log{(n/d)})$. When $d$, or at least some non-trivial estimate of $d$, is known, we prove a tight bound of $\Theta(d\log (n/d))$ for the deterministic non-adaptive settings, and $\Theta(\ell\log(n/d))$ for the randomized non-adaptive settings. In the adaptive case, we present an upper bound of $O(\ell \log (n/\ell))$ for the deterministic settings, and a lower bound of $\Omega(\ell\log(n/d)+\log n)$. Additionally, we establish a tight bound of $\Theta(\ell \log(n/d))$ for the randomized adaptive settings

Improved Lower Bound for Estimating the Number of Defective Items

Let $X$ be a set of items of size $n$ that contains some defective items, denoted by $I$, where $I \subseteq X$. In group testing, a {\it test} refers to a subset of items $Q \subset X$. The outcome of a test is $1$ if $Q$ contains at least one defective item, i.e., $Q\cap I \neq \emptyset$, and $0$ otherwise. We give a novel approach to obtaining lower bounds in non-adaptive randomized group testing. The technique produced lower bounds that are within a factor of $1/{\log\log\stackrel{k}{\cdots}\log n}$ of the existing upper bounds for any constant~$k$. Employing this new method, we can prove the following result. For any fixed constants $k$, any non-adaptive randomized algorithm that, for any set of defective items $I$, with probability at least $2/3$, returns an estimate of the number of defective items $|I|$ to within a constant factor requires at least $$\Omega\left(\frac{\log n}{\log\log\stackrel{k}{\cdots}\log n}\right)$$ tests. Our result almost matches the upper bound of $O(\log n)$ and solves the open problem posed by Damaschke and Sheikh Muhammad [COCOA 2010 and Discrete Math., Alg. and Appl., 2010]. Additionally, it improves upon the lower bound of $\Omega(\log n/\log\log n)$ previously established by Bshouty [ISAAC 2019]

Efficiently Decodable Non-Adaptive Threshold Group Testing

We consider non-adaptive threshold group testing for identification of up to $d$ defective items in a set of $n$ items, where a test is positive if it contains at least $2 \leq u \leq d$ defective items, and negative otherwise. The defective items can be identified using $t = O \left( \left( \frac{d}{u} \right)^u \left( \frac{d}{d - u} \right)^{d-u} \left(u \log{\frac{d}{u}} + \log{\frac{1}{\epsilon}} \right) \cdot d^2 \log{n} \right)$ tests with probability at least $1 - \epsilon$ for any $\epsilon > 0$ or $t = O \left( \left( \frac{d}{u} \right)^u \left( \frac{d}{d -u} \right)^{d - u} d^3 \log{n} \cdot \log{\frac{n}{d}} \right)$ tests with probability 1. The decoding time is $t \times \mathrm{poly}(d^2 \log{n})$. This result significantly improves the best known results for decoding non-adaptive threshold group testing: $O(n\log{n} + n \log{\frac{1}{\epsilon}})$ for probabilistic decoding, where $\epsilon > 0$, and $O(n^u \log{n})$ for deterministic decoding