Improved Lower Bounds for Strongly Separable Matrices and Related Combinatorial Structures

TL;DR

This paper improves the lower bounds on the rate of strongly separable matrices, which are crucial in nonadaptive group testing, by employing a modified probabilistic method, and extends these improvements to related combinatorial structures.

Contribution

It introduces a modified probabilistic approach that enhances lower bounds for strongly separable matrices and related combinatorial structures.

Findings

Improved lower bounds for the rate of strongly separable matrices.

Enhanced bounds for locally thin set families.

Better bounds for cancellative set families.

Abstract

In nonadaptive group testing, the main research objective is to design an efficient algorithm to identify a set of up to $t$ positive elements among $n$ samples with as few tests as possible. Disjunct matrices and separable matrices are two classical combinatorial structures while one provides a more efficient decoding algorithm and the other needs fewer tests, i.e., larger rate. Recently, a notion of strongly separable matrix has been introduced, which has the same identifying ability as a disjunct matrix, but has larger rate. In this paper, we use a modified probabilistic method to improve the lower bounds for the rate of strongly separable matrices. Using this method, we also improve the lower bounds for some well-known combinatorial structures, including locally thin set families and cancellative set families.