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A Unified View of Graph Regularity via Matrix Decompositions

Greg Bodwin, Santosh Vempala

TL;DR

The paper introduces cut pseudorandom graphs and a matrix-decomposition framework to unify sparse graph regularity results. By a deterministic, polynomial-time cut decomposition process, it defines a class of graphs on which both sparse weak regularity and sparse Szemerédi-type regularity hold, with bounds tied to projection-values of the decomposition. It then shows that several natural sparse graph classes (L^p upper regular, low threshold rank, and core-dense) are encompassed by cut pseudorandomness, yielding new regularity results and enabling PTASes for MAX-CUT, MAX-BISECTION, and MIN-BISECTION; the approach also extends to tensor decompositions for MAX-CSP. The framework thus generalizes and strengthens the linear-algebraic foundations of sparse regularity, providing both algorithmic guarantees and broad applicability to combinatorial optimization on sparse graphs. This unified perspective offers a path to efficient approximation schemes in settings where classical dense-graph regularity is inapplicable, with practical impact on graph algorithms and CSPs in sparse regimes.

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.)

A Unified View of Graph Regularity via Matrix Decompositions

TL;DR

The paper introduces cut pseudorandom graphs and a matrix-decomposition framework to unify sparse graph regularity results. By a deterministic, polynomial-time cut decomposition process, it defines a class of graphs on which both sparse weak regularity and sparse Szemerédi-type regularity hold, with bounds tied to projection-values of the decomposition. It then shows that several natural sparse graph classes (L^p upper regular, low threshold rank, and core-dense) are encompassed by cut pseudorandomness, yielding new regularity results and enabling PTASes for MAX-CUT, MAX-BISECTION, and MIN-BISECTION; the approach also extends to tensor decompositions for MAX-CSP. The framework thus generalizes and strengthens the linear-algebraic foundations of sparse regularity, providing both algorithmic guarantees and broad applicability to combinatorial optimization on sparse graphs. This unified perspective offers a path to efficient approximation schemes in settings where classical dense-graph regularity is inapplicable, with practical impact on graph algorithms and CSPs in sparse regimes.

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) 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.)

Paper Structure

This paper contains 24 sections, 19 theorems, 74 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Cut Decompositions
  3. Regularity for Cut Pseudorandom Graphs
  4. Graph Classes Captured by Cut Pseudorandomness
  5. Algorithmic Applications
  6. Cut Decompositions
  7. Basic Cut Decompositions
  8. Decomposition Algorithm.
  9. Definitions and Properties.
  10. Cut Decompositions under Diagonal Inner Products
  11. Projection Values vs. Matrix Norms
  12. Sparse Regularity Lemmas for Cut Pseudorandom Graphs
  13. Sparse Cut Approximation and Sparse Weak Regularity
  14. Sparse Szemerédi Regularity
  15. The First Norm.
  16. ...and 9 more sections

Key Result

Theorem 1

For any binary matrix $A \in \mathbb{R}^{n \times n}$ and $\varepsilon > 0$, there is $\widehat{A}$ that is a sum of $O(\varepsilon^{-2})$ cut matrices such that

Figures (1)

  • Figure 1: Some existing implications in the area of graph regularity (top), and how the results in this paper change the picture (bottom). For clarity we have omitted arrows in these diagrams that are implied transitively by a chain of other arrows, even when these direct implications may have advantages (e.g. better quantitative bounds or faster computation). Citations and explanations are given in the text below.

Theorems & Definitions (40)

  • Theorem 1: Cut Approximation FK99
  • Theorem 2: Weak Regularity Lemma FK99
  • Theorem 3: Szemerédi Regularity Lemma Szemeredi75LS07
  • Definition 1: Cut Pseudorandomness
  • Theorem 4: Regularity Lemmas for Cut Pseudorandom Graphs
  • Definition 2: $L^p$ Upper Regularity BCCZ14BCCZ14b
  • Definition 3: Low Threshold Rank BRS11GS11GS12GS13GharanT15
  • Definition 4: Core Density KV09
  • Theorem 5: Cut Pseudorandomness of Graph Classes
  • Corollary 6
  • ...and 30 more