A Unified View of Graph Regularity via Matrix Decompositions

TL;DR

This paper establishes new regularity lemmas for sparse graphs using an abstract matrix decomposition approach, unifying various graph classes and enabling improved approximation algorithms for combinatorial problems.

Contribution

It introduces the concept of cut pseudorandom graphs and proves regularity lemmas for them, unifying multiple graph classes under a common framework.

Findings

Proves regularity lemmas for core-dense, low threshold rank, and $L^p$ upper regular graphs.

Defines cut pseudorandom graphs capturing these classes as special cases.

Enables new PTASes for MAX-CUT, MAX-BISECTION, MIN-BISECTION on a broad class of graphs.

Abstract

We prove algorithmic weak and \Szemeredi{} regularity lemmas for several classes of sparse graphs in the literature, for which only weak regularity lemmas were previously known. These include core-dense graphs, low threshold rank graphs, and (a version of) $L^p$ upper regular graphs. More precisely, we define \emph{cut pseudorandom graphs}, we prove our regularity lemmas for these graphs, and then we show that cut pseudorandomness captures all of the above graph classes as special cases. The core of our approach is an abstracted matrix decomposition, roughly following Frieze and Kannan [Combinatorica '99] and \Lovasz{} and Szegedy [Geom.\ Func.\ Anal.\ '07], which can be computed by a simple algorithm by Charikar [AAC0 '00]. This gives rise to the class of cut pseudorandom graphs, and using work of Oveis Gharan and Trevisan [TOC '15], it also implies new PTASes for MAX-CUT, MAX-BISECTION, MIN-BISECTION for a significantly expanded class of input graphs. (It is NP Hard to get PTASes for these graphs in general.)