Efficient Removal Lemmas for Matrices

TL;DR

This paper develops more efficient removal lemmas for specific cases of hereditary matrix properties, reducing the complexity of detecting forbidden submatrices in large matrices.

Contribution

It establishes polynomial bounds for removal lemmas in special matrix cases, improving upon previous exponential bounds and advancing towards a conjecture by Alon, Fischer, and Newman.

Findings

Polynomial bounds for removal lemmas in fixed submatrix cases

Extension of regularity lemma techniques to new combinatorial settings

Progress towards proving a conjecture on matrix property testing

Abstract

The authors and Fischer recently proved that any hereditary property of two-dimensional matrices (where the row and column order is not ignored) over a finite alphabet is testable with a constant number of queries, by establishing the following (ordered) matrix removal lemma: For any finite alphabet $\Sigma$, any hereditary property $\mathcal{P}$ of matrices over $\Sigma$, and any $\epsilon > 0$, there exists $f_{\mathcal{P}}(\epsilon)$ such that for any matrix $M$ over $\Sigma$ that is $\epsilon$-far from satisfying $\mathcal{P}$, most of the $f_{\mathcal{P}}(\epsilon) \times f_{\mathcal{P}}(\epsilon)$ submatrices of $M$ do not satisfy $\mathcal{P}$. Here being $\epsilon$-far from $\mathcal{P}$ means that one needs to modify at least an $\epsilon$-fraction of the entries of $M$ to make it satisfy $\mathcal{P}$. However, in the above general removal lemma, $f_{\mathcal{P}}(\epsilon)$ grows very fast as a function of $\epsilon^{-1}$, even when $\mathcal{P}$ is characterized by a single forbidden submatrix. In this work we establish much more efficient removal lemmas for several special cases of the above problem. In particular, we show the following: For any fixed $s \times t$ binary matrix $A$ and any $\epsilon > 0$ there exists $\delta > 0$ polynomial in $\epsilon$, such that for any binary matrix $M$ in which less than a $\delta$-fraction of the $s \times t$ submatrices are equal to $A$, there exists a set of less than an $\epsilon$-fraction of the entries of $M$ that intersects every $A$-copy in $M$. We generalize the work of Alon, Fischer and Newman [SICOMP'07] and make progress towards proving one of their conjectures. The proofs combine their efficient conditional regularity lemma for matrices with additional combinatorial and probabilistic ideas.