Almost Separable Matrices

TL;DR

This paper introduces the concept of almost $k$-separable matrices, showing they can be constructed with near-optimal row counts and providing new bounds for nonadaptive group testing.

Contribution

It defines almost $k$-separability, proves their existence with optimal row counts, and applies findings to improve bounds in nonadaptive group testing.

Findings

Existence of almost $k$-separable matrices with $O(k \, \log n)$ rows.

Optimality of these matrices for certain $k$ ranges.

New bounds on the rate of nonadaptive group testing.

Abstract

An $m \times n$ matrix $\mathsf{A}$ with column supports $\{S_i\}$ is $k$-separable if the disjunctions $\bigcup_{i \in \mathcal{K}} S_i$ are all distinct over all sets $\mathcal{K}$ of cardinality $k$. While a simple counting bound shows that $m > k \log_2 n/k$ rows are required for a separable matrix to exist, in fact it is necessary for $m$ to be about a factor of $k$ more than this. In this paper, we consider a weaker definition of `almost $k$-separability', which requires that the disjunctions are `mostly distinct'. We show using a random construction that these matrices exist with $m = O(k \log n)$ rows, which is optimal for $k = O(n^{1-\beta})$. Further, by calculating explicit constants, we show how almost separable matrices give new bounds on the rate of nonadaptive group testing.