An algorithmic regularity lemma for $L_p$ regular sparse matrices

TL;DR

This paper introduces an algorithmic regularity lemma for $L_p$ regular sparse matrices, extending previous work for $L_{ ext{infinity}}$ matrices, with applications to tensors and MAX-CSP problems.

Contribution

The paper extends the regularity lemma to $L_p$ regular matrices for $1 < p \,\leq\, \infty$, broadening its applicability to sparse matrices.

Findings

Established an algorithmic regularity lemma for $L_p$ regular matrices.

Extended previous results from $L_{\infty}$ to general $L_p$ cases.

Applied the lemma to tensors and MAX-CSP instances.

Abstract

We prove an algorithmic regularity lemma for $L_p$ regular matrices $(1 < p \leq \infty),$ a class of sparse $\{0,1\}$ matrices which obey a natural pseudorandomness condition. This extends a result of Coja-Oghlan, Cooper and Frieze who treated the case of $L_{\infty}$ regular matrices. We also present applications of this result for tensors and MAX-CSP instances.