Group Testing Problem Research Articles

Adaptive Group Testing on Networks With Community Structure: The Stochastic Block Model

Group testing was conceived during World War II to identify soldiers infected with syphilis using as few tests as possible, and it has attracted renewed interest during the COVID-19 pandemic. A long-standing assumption in the probabilistic variant of the group testing problem is that individuals are infected by the disease <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">independently</i> . However, this assumption rarely holds in practice, as diseases often spread through interactions between individuals and therefore cause infections to be correlated. Inspired by characteristics of COVID-19 and other infectious diseases, we introduce an infection model over networks which generalizes the traditional i.i.d. model from probabilistic group testing. Under this model, we ask whether knowledge of the network structure can be leveraged to perform group testing more efficiently, focusing specifically on community-structured graphs drawn from the stochastic block model. We prove that a simple community-aware algorithm outperforms the baseline binary splitting algorithm when the model parameters are conducive to “strong community structure.” Moreover, our novel lower bounds imply that the community-aware algorithm is order-optimal in certain parameter regimes. We extend our bounds to the noisy setting and support our results with numerical experiments.

Upper and lower bounds for competitive group testing

We consider competitive algorithms for adaptive group testing problems. In the first part of the paper, we develop an algorithm with competitive constant c < 1.452 thus improving the up to now best known algorithms with constants 1.5+epsilon from 2003. In the second part, we prove the first nontrivial lower bound for competitive constants, namely that c is always larger than 1.31.

Random Multi-access Scheme Based on Group Testing Theory for Massive Machine-type Communication

Considering emerging massive machine-type communication with very few devices simultaneously transmitting short messages, random multiple access has currently become a research hot spot since the traditional multiple access protocol struggles to work properly. Inspired by the superimposed code, a channel assignment scheme based on non-adaptive group testing theory is proposed to allow multiple users to share a single channel at the same time. A superimposed code with threshold value d, which can recover the original code through the Boolean sum of up to d codes, is constructed by studying the problem of group testing. When the number of active users with a message to be transmitted is no more than the threshold value, active users encode their messages by the superimposed code and multiple active users transmit their messages together by sharing a single channel. The receiver can decode the mixed messages by the disjunction of the superimposed code. We focus on the design of the superimposed code by group testing theory, formulate this problem as the construction of a transversal design, and use the Latin square to form a useful transversal design. Theoretical proof shows that our design works well under limited circumstances where the number of active users does not exceed the threshold value. Additionally, our design can recover the original codes by the mixture of codes, which are more easily than the algorithms based on compressive sensing. Simulation analysis presents that our design performs better than the traditional competitive access protocols in terms of successful probability, throughput, and packet loss rate.

Performance Bounds for Group Testing With Doubly-Regular Designs

In the group testing problem, the goal is to identify a subset of defective items within a larger set of items based on tests whose outcomes indicate whether any defective item is present. This problem is relevant in areas such as medical testing, DNA sequencing, and communications. In this paper, we study a doubly-regular design in which the number of tests-per-item and the number of items-per-test are fixed. We analyze the performance of this test design alongside the Definite Defectives (DD) decoding algorithm in several settings, namely, (i) the sub-linear regime <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$k=o(n)$ </tex-math></inline-formula> with exact recovery, (ii) the linear regime <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$k=\Theta (n)$ </tex-math></inline-formula> with approximate recovery, and (iii) the size-constrained setting, where the number of items per test is constrained. Under setting (i), we show that our design together with the DD algorithm, matches an existing achievability result for the DD algorithm with the near-constant tests-per-item design, which is known to be asymptotically optimal in broad scaling regimes. Under setting (ii), we provide novel approximate recovery bounds that complement a hardness result regarding exact recovery. Lastly, under setting (iii), we improve on the best known upper and lower bounds in scaling regimes where the maximum test size grows with the total number of items.

Dynamic Infection Spread Model Based Group Testing

Group testing idea is an efficient approach to detect prevalence of an infection in the test samples taken from a group of individuals. It is based on the idea of pooling the test samples and performing tests to the mixed samples. This approach results in possible reduction in the required number of tests to identify infections. Classical group testing works consider static settings where the infection statuses of the individuals do not change throughout the testing process. In our paper, we study a dynamic infection spread model, inspired by the discrete time SIR model, where infections are spread via non-isolated infected individuals, while infection keeps spreading over time, a limited capacity testing is performed at each time instance as well. In contrast to the classical, static group testing problem, the objective in our setup is not to find the minimum number of required tests to identify the infection status of every individual in the population, but to control the infection spread by detecting and isolating the infections over time by using the given, limited number of tests. In order to analyze the performance of the proposed algorithms, we focus on the average-case analysis of the number of individuals that remain non-infected throughout the process of controlling the infection. We propose two dynamic algorithms that both use given limited number of tests to identify and isolate the infections over time, while the infection spreads, while the first algorithm is a dynamic randomized individual testing algorithm, in the second algorithm we employ the group testing approach similar to the original work of Dorfman. By considering weak versions of our algorithms, we obtain lower bounds for the performance of our algorithms. Finally, we implement our algorithms and run simulations to gather numerical results and compare our algorithms and theoretical approximation results under different sets of system parameters.

A negative binomial approximation in group testing

AbstractWe consider the problem of group testing (pooled testing), first introduced by Dorfman. For nonadaptive testing strategies, we refer to a nondefective item as “intruding” if it only appears in positive tests. Such items cause misclassification errors in the well-known COMP algorithm and can make other algorithms produce an error. It is therefore of interest to understand the distribution of the number of intruding items. We show that, under Bernoulli matrix designs, this distribution is well approximated in a variety of senses by a negative binomial distribution, allowing us to understand the performance of the two-stage conservative group testing algorithm of Aldridge.

On the Binary Adder Channel With Complete Feedback, With an Application to Quantitative Group Testing

We determine the exact value of the optimal symmetric rate point (r, r) in the Dueck zero-error capacity region of the binary adder channel with complete feedback. We proved that the average zero-error capacity r = h(1/2-δ) ≈ 0.78974, where h(∙) is the binary entropy function and δ = 1/(2 log2(2 + √3)). Our motivation is a problem in quantitative group testing. Given a set of n elements two of which are defective, the quantitative group testing problem asks for the identification of these two defectives through a series of tests. Each test gives the number of defectives contained in the tested subset, and the outcomes of previous tests are assumed known at the time of designing the current test. We establish that the minimum number of tests is asymptotic to (log2 n)/r as n → ∞.

Noisy Adaptive Group Testing via Noisy Binary Search

The group testing problem consists of determining a small set of defective items from a larger set of items based on a number of possibly-noisy tests, and has numerous practical applications. One of the defining features of group testing is whether the tests are adaptive (i.e., a given test can be chosen based on all previous outcomes) or non-adaptive (i.e., all tests must be chosen in advance). In this paper, building on the success of binary splitting techniques in noiseless group testing (Hwang, 1972), we introduce noisy group testing algorithms that apply noisy binary search as a subroutine. We provide three variations of this approach with increasing complexity, culminating in an algorithm that succeeds using a number of tests that matches the best known previously (Scarlett, 2019), while overcoming fundamental practical limitations of the existing approach, and more precisely capturing the dependence of the number of tests on the error probability. We provide numerical experiments demonstrating that adaptive group testing strategies based on noisy binary search can be highly effective in practice, using significantly fewer tests compared to state-of-the-art non-adaptive strategies.

Improved Bounds for Noisy Group Testing With Constant Tests per Item

The group testing problem is concerned with identifying a small set of infected individuals in a large population. At our disposal is a testing procedure that allows us to test several individuals together. In an idealized setting, a test is positive if and only if at least one infected individual is included and negative otherwise. Significant progress was made in recent years towards understanding the information-theoretic and algorithmic properties in this noiseless setting. In this paper, we consider a noisy variant of group testing where test results are flipped with certain probability, including the realistic scenario where sensitivity and specificity can take arbitrary values. Using a test design where each individual is assigned to a fixed number of tests, we derive explicit algorithmic bounds for two commonly considered inference algorithms and thereby naturally extend the results of Scarlett & Cevher (2016) and Scarlett & Johnson (2020). We provide improved performance guarantees for the efficient algorithms in these noisy group testing models – indeed, for a large set of parameter choices the bounds provided in the paper are the strongest currently proved.

Generalized framework for Group Testing: Queries, feedbacks and adversaries

In the Group Testing problem, the objective is to learn a subset K of some much larger domain N, using the shortest-possible sequence of queries Q. A feedback to a query provides some information about the intersection between the query and subset K. Several specific feedbacks have been studied in the literature, often proving different formulas for the estimate of the query complexity of the problem, defined as the shortest length of queries' sequence solving Group Testing problem with specific feedback. In this paper we study what are the properties of the feedback that influence the query complexity of Group Testing and what is their measurable impact. We propose a generic framework that covers a vast majority of relevant settings considered in the literature, which depends on two fundamental parameters of the feedback: input capacity α and output expressiveness β. They upper bound the logarithm of the size of the feedback function domain and image, respectively. To justify the value of the framework, we prove upper bounds on query complexity of non-adaptive, deterministic Group Testing under some “efficient” feedbacks, for minimum, maximum and general expressiveness, and complement them with a lower bound on all feedbacks with given parameters α,β. Our upper bounds also hold if the feedback function could get an input twisted by a malicious adversary, in case the intersection of a query and the hidden set is bigger than the feedback capacity α. We also show that slight change in the feedback function may result in substantial worsening of the query complexity. Additionally, we analyze explicitly constructed randomized counterparts of the deterministic results. Our results provide some insights to what are the most useful bits of information an output-restricted feedback could provide, and open a number of challenging research directions.

Random and quasi-random designs in group testing

For large classes of group testing problems, we derive lower bounds for the probability that all significant items are uniquely identified using specially constructed random designs. These bounds allow us to optimize parameters of the randomization schemes. We also suggest and numerically justify a procedure of constructing designs with better separability properties than pure random designs. We illustrate theoretical considerations with a large simulation-based study. This study indicates, in particular, that in the case of the common binary group testing, the suggested families of designs have better separability than the popular designs constructed from disjunct matrices. We also derive several asymptotic expansions and discuss the situations when the resulting approximations achieve high accuracy.

Optimal non-adaptive probabilistic group testing in general sparsity regimes

Abstract In this paper, we consider the problem of noiseless non-adaptive probabilistic group testing, in which the goal is high-probability recovery of the defective set. We show that in the case of $n$ items among which $k$ are defective, the smallest possible number of tests equals $\min \{ C_{k,n} k \log n, n\}$ up to lower-order asymptotic terms, where $C_{k,n}$ is a uniformly bounded constant (varying depending on the scaling of $k$ with respect to $n$) with a simple explicit expression. The algorithmic upper bound follows from a minor adaptation of an existing analysis of the Definite Defectives algorithm, and the algorithm-independent lower bound builds on existing works for the regimes $k \le n^{1-\varOmega (1)}$ and $k = \varTheta (n)$. In sufficiently sparse regimes (including $k = o\big ( \frac{n}{\log n} \big )$), our main result generalizes that of Coja-Oghlan et al. (2020) by avoiding the assumption $k \le n^{1-\varOmega (1)}$, whereas in sufficiently dense regimes (including $k = \omega \big ( \frac{n}{\log n} \big )$), our main result shows that individual testing is asymptotically optimal for any non-zero target success probability, thus strengthening an existing result of Aldridge (2019, IEEE Trans. Inf. Theory, 65, 2058–2061) in terms of both the error probability and the assumed scaling of $k$.

A tractable non-adaptative group testing method for non-binary measurements

The original problem of group testing consists in the identification of defective items in a collection, by applying tests on groups of items that detect the presence of at least one defective element in the group. The aim is then to identify all defective items of the collection with as few tests as possible. This problem is relevant in several fields, among which biology and computer sciences. In the present article we consider that the tests applied to groups of items returns a load, measuring how defective the most defective item of the group is. In this setting, we propose a simple non-adaptative algorithm allowing the detection of all defective items of the collection. Items are put on an n × n grid and pools are organised as lines, columns and diagonals of this grid. This method improves on classical group testing algorithms using only the binary response of the test. Group testing recently gained attraction as a potential tool to solve a shortage of COVID-19 test kits, in particular for RT-qPCR. These tests return the viral load of the sample and the viral load varies greatly among individuals. Therefore our model presents some of the key features of this problem. We aim at using the extra piece of information that represents the viral load to construct a one-stage pool testing algorithm on this idealized version. We show that under the right conditions, the total number of tests needed to detect contaminated samples can be drastically diminished.

Model-Based and Graph-Based Priors for Group Testing

The goal of the group testing problem is to identify a set of defective items within a larger set of items, using suitably-designed tests whose outcomes indicate whether any defective item is present. In this paper, we study how the number of tests can be significantly decreased by leveraging the structural dependencies between the items, i.e., by incorporating prior information. To do so, we pursue two different perspectives: (i) As a generalization of the uniform combinatorial prior, we consider the case that the defective set is uniform over a \emph{subset} of all possible sets of a given size, and study how this impacts the information-theoretic limits on the number of tests for approximate recovery; (ii) As a generalization of the i.i.d.~prior, we introduce a new class of priors based on the Ising model, where the associated graph represents interactions between items. We show that this naturally leads to an Integer Quadratic Program decoder, which can be converted to an Integer Linear Program and/or relaxed to a non-integer variant for improved computational complexity, while maintaining strong empirical recovery performance.

Optimal group testing

AbstractIn the group testing problem the aim is to identify a small set of k ⁓ nθ infected individuals out of a population size n, 0 &lt; θ &lt; 1. We avail ourselves of a test procedure capable of testing groups of individuals, with the test returning a positive result if and only if at least one individual in the group is infected. The aim is to devise a test design with as few tests as possible so that the set of infected individuals can be identified correctly with high probability. We establish an explicit sharp information-theoretic/algorithmic phase transition minf for non-adaptive group testing, where all tests are conducted in parallel. Thus with more than minf tests the infected individuals can be identified in polynomial time with high probability, while learning the set of infected individuals is information-theoretically impossible with fewer tests. In addition, we develop an optimal adaptive scheme where the tests are conducted in two stages.

Sublinear-Time Non-Adaptive Group Testing With O(k log n) Tests via Bit-Mixing Coding

The group testing problem consists of determining a small set of defective items from a larger set of items based on tests on groups of items, and is relevant in applications such as medical testing, communication protocols, pattern matching, and many more. While rigorous group testing algorithms have long been known with runtime at least linear in the number of items, a recent line of works has sought to reduce the runtime to ${\mathrm{ poly}}({k} \log {n})$ , where n is the number of items and k is the number of defectives. In this paper, we present such an algorithm for non-adaptive group testing termed bit mixing coding (BMC), which builds on techniques that encode item indices in the test matrix, while incorporating novel ideas based on erasure-correction coding. We show that BMC achieves asymptotically vanishing error probability with ${O}({k} \log {n})$ tests and ${O}({k}^{2} \cdot \log {k} \cdot \log {n})$ runtime, in the limit as ${n} \to \infty $ (with k having an arbitrary dependence on n). This closes an open problem of simultaneously achieving ${\mathrm{ poly}}({k} \log {n})$ decoding time using ${O}({k} \log {n})$ tests without any assumptions on k. In addition, we show that the same scaling laws can be attained in a commonly-considered noisy setting, in which each test outcome is flipped with constant probability.

Information-Theoretic and Algorithmic Thresholds for Group Testing

In the group testing problem we aim to identify a small number of infected individuals within a large population. We avail ourselves to a procedure that can test a group of multiple individuals, with the test result coming out positive iff at least one individual in the group is infected. With all tests conducted in parallel, what is the least number of tests required to identify the status of all individuals? In a recent test design [Aldridge et al.\ 2016] the individuals are assigned to test groups randomly, with every individual joining an equal number of groups. We pinpoint the sharp threshold for the number of tests required in this randomised design so that it is information-theoretically possible to infer the infection status of every individual. Moreover, we analyse two efficient inference algorithms. These results settle conjectures from [Aldridge et al.\ 2014, Johnson et al.\ 2019].

New construction of error-correcting pooling designs from singular linear spaces over finite fields

The group testing problem is that we are asked to identify all the defects with the minimum number of tests when given a set of n items with at most d defects. In this paper, as a generalization of Liu et al.’s construction in the paper (Liu and Gao in Discret Math 338:857–862, 2015), new pooling designs are constructed from singular linear spaces over finite fields. Then we make comparisons with Liu et al.’s construction in the aspects of parameters of pooling designs. By choosing appropriate parameters in our pooling designs, the performance of test efficiency in our pooling designs is better than that given by Liu et al. Finally, the analysis of parameters in our pooling designs is provided.

On optimal randomized group testing with one defective item and a constrained number of positive responses

Consider the group testing problem with an input set of size n and the number of defective items being d=1, that allows at most y positive responses. Let An,y denote the minimum expected number of tests required by a randomized testing strategy for this problem. The testing strategy is required to be Las Vagas, that is, guaranteed to give the correct classification of the items. Based on Yao’s minimax principle, An,y is equal to the minimum average number of tests for a deterministic group testing strategy allowing at most y positive responses, where the average is taken over all the n possible input sets of size n with d=1.The main contribution of our paper is to show that An,y is yy+1(ny!)1y+O(y), for 1≤y<⌈log2n⌉. This solves an open question left in Damaschke (2016).

On the All-or-Nothing Behavior of Bernoulli Group Testing

In this article, we study the problem of non-adaptive group testing, in which one seeks to identify which items are defective given a set of suitably-designed tests whose outcomes indicate whether or not at least one defective item was included in the test. The most widespread recovery criterion seeks to exactly recover the entire defective set, and relaxed criteria such as approximate recovery and list decoding have also been considered. In contrast, we study the fundamental limits of group testing under the significantly relaxed weak recovery criterion, which only seeks to identify a small fraction (e.g., 0.01) of the defective items. Given the near-optimality of i.i.d. Bernoulli testing for exact recovery in sufficiently sparse scaling regimes, it is natural to ask whether this design additionally succeeds with much fewer tests under weak recovery. Our main negative result shows that this is not the case, and in fact, under i.i.d. Bernoulli random testing in the sufficiently sparse regime, an all-or-nothing phenomenon occurs: When the number of tests is slightly below a threshold, weak recovery is impossible, whereas when the number of tests is slightly above the same threshold, high-probability exact recovery is possible. In establishing this result, we additionally prove similar negative results under Bernoulli designs for the weak detection problem (distinguishing between the group testing model vs. completely random outcomes) and the problem of identifying a single item that is definitely defective. On the positive side, we show that all three relaxed recovery criteria can be attained using considerably fewer tests under suitably-chosen non-Bernoulli designs. Thus, our results collectively indicate that when too few tests are available, naively applying i.i.d. Bernoulli testing can lead to catastrophic failure, whereas “cutting one’s losses” and adopting a more carefully-chosen design can still succeed in attaining these less stringent criteria.