Nearly Optimal Sparse Group Testing

TL;DR

This paper investigates sparse group testing models with physical constraints, providing theoretical bounds and efficient constructions for testing schemes that minimize tests while ensuring accurate identification of defective items.

Contribution

It introduces new information-theoretic bounds and explicit, computationally efficient designs for sparse group testing under physical constraints.

Findings

Lower bounds on the number of tests for sparse models

Explicit constructions with near-optimal test counts

Universal randomized design for size-constrained tests

Abstract

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.