Sparse Combinatorial Group Testing

TL;DR

This paper investigates the sparse regime of combinatorial group testing where each item or test is limited in participation, providing new bounds and constructions that drastically reduce the number of tests needed, especially when constraints are slightly above minimal levels.

Contribution

The paper introduces a modified Kautz-Singleton construction for sparse group testing, characterizes the number of tests as a function of participation limits, and demonstrates near order-optimality in this regime.

Findings

Number of tests decreases sharply when item participation limit exceeds the minimum.

For w_max ≤ d, individual testing is optimal with t=n.

Modified Kautz-Singleton construction achieves near order-optimal tests for larger participation limits.

Abstract

In combinatorial group testing (CGT), the objective is to identify the set of at most $d$ defective items from a pool of $n$ items using as few tests as possible. The celebrated result for the CGT problem is that the number of tests $t$ can be made logarithmic in $n$ when $d=O(poly(\log n))$. However, state-of-the-art GT codes require the items to be tested $w=\Omega(d\log n)$ times and tests to include $\rho=\Omega(n/d)$ items (within log factors). In many applications, items can only participate in a limited number of tests and tests are constrained to include a limited number of items. In this paper, we study the "sparse" regime for the group testing problem where we restrict the number of tests each item can participate in by $w_{\max}$ or the number of items each test can include by $\rho_{\max}$ in both noiseless and noisy settings. These constraints lead to an unexplored regime where $t$ is a fractional power of $n$. Our results characterize the number of tests $t$ as a function of $w_{\max} (\rho_{\max})$ and show, for example, that $t$ decreases drastically when $w_{\max}$ is increased beyond a bare minimum. In particular, if $w_{\max}\leq d$, then we must have $t=n$, i.e., individual testing is optimal. We show that if $w_{\max}=d+1$, this decreases suddenly to $t=\Theta(d\sqrt{n})$. The order-optimal construction is obtained via a modification of the Kautz-Singleton construction, which is known to be suboptimal for the classical GT problem. For more general case, when $w_{\max}=ld+1$ for $l>1$, the modified K-S construction requires $t=\Theta(d n^{\frac{1}{l+1}})$ tests, which we prove to be near order-optimal. We show that our constructions have a favorable encoding and decoding complexity. We finally discuss an application of our results to the construction of energy-limited random access schemes for IoT networks, which provided the initial motivation for our work.