Property testing of the Boolean and binary rank

TL;DR

This paper introduces algorithms for efficiently testing whether a 0-1 matrix has Boolean or binary rank at most d, with query complexities depending on d and epsilon, enabling quick property verification.

Contribution

It provides the first known property testing algorithms for Boolean and binary rank of matrices with specific query complexities.

Findings

Boolean rank testing query complexity: (d^4/^6)

Binary rank testing query complexity: O(2^{2d}/)

Algorithms are one-sided error, always accepting low-rank matrices and rejecting ar matrices with high probability.

Abstract

We present algorithms for testing if a $(0,1)$-matrix $M$ has Boolean/binary rank at most $d$, or is $\epsilon$-far from Boolean/binary rank $d$ (i.e., at least an $\epsilon$-fraction of the entries in $M$ must be modified so that it has rank at most $d$). The query complexity of our testing algorithm for the Boolean rank is $\tilde{O}\left(d^4/ \epsilon^6\right)$. For the binary rank we present a testing algorithm whose query complexity is $O(2^{2d}/\epsilon)$. Both algorithms are $1$-sided error algorithms that always accept $M$ if it has Boolean/binary rank at most $d$, and reject with probability at least $2/3$ if $M$ is $\epsilon$-far from Boolean/binary rank $d$.