Combinatorial Group Testing and Sparse Recovery Schemes with Near-Optimal Decoding Time

In the long-studied problem of combinatorial group testing, one is asked to detect a set of $k$ defective items out of a population of size $n$, using $m\ll n$ disjunctive measurements. In the non-adaptive setting, the most widely used combinatorial objects are disjunct and list-disjunct matrices, which define incidence matrices of test schemes. Disjunct matrices allow the identification of the exact set of defectives, whereas list disjunct matrices identify a small superset of the defectives. Apart from the combinatorial guarantees, it is often of key interest to equip measurement designs with efficient decoding algorithms. The most efficient decoders should run in sublinear time in $n$, and ideally near-linear in the number of measurements $m$. In this work, we give several constructions with an optimal number of measurements and near-optimal decoding time for the most fundamental group testing tasks, as well as for central tasks in the compressed sensing and heavy hitters literature. For many of those tasks, the previous measurement-optimal constructions needed time either quadratic in the number of measurements or linear in the universe size. Among our results are the following: a construction of disjunct matrices matching the best-known construction in terms of the number of rows $m$, but achieving nearly linear decoding time in $m$; a construction of list disjunct matrices with the optimal $m=O(k\log(n/k)$ number of rows and nearly linear decoding time in $m$; error-tolerant variations of the above constructions; a non-adaptive group testing scheme for the “for-each” model with $m=O(k\log n)$ measurements and $O(m)$ decoding time; a streaming algorithm for the “for-all” version of the heavy hitters problem in the strict turnstile model with near-optimal query time, as well as a “list decoding” variant obtaining also near-optimal update time and $O(k\log(n/k))$ space usage; an $\ell_{2}/\ell_{2}$ weak identification system for compressed sensing with nearly optimal sample complexity and nearly linear decoding time in the sketch length. Most of our results are obtained via a clean and novel approach that avoids list-recoverable codes or related complex techniques that were present in almost every state-of-the-art work on efficiently decodable constructions of such objects.