Concomitant Group Testing

TL;DR

This paper introduces Concomitant Group Testing, a new problem where tests require items from multiple semi-defective sets, and provides algorithms that are order-optimal for identifying these sets efficiently.

Contribution

The paper formulates Concomitant Group Testing, develops algorithms for the case of two semi-defective sets, and proves their order-optimality under various adaptivity constraints.

Findings

Deterministic adaptive algorithm is order-optimal.

Randomized algorithms are order-optimal with limited adaptivity.

Significant improvements over baseline hypergraph learning methods.

Abstract

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).