A Cryptographic View of Regularity Lemmas: Simpler Unified Proofs and Refined Bounds

Abstract

In this work we present a short and unified proof for the Strong and Weak Regularity Lemma, based on the cryptographic technique called low-complexity approximations. In short, both problems reduce to a task of finding constructively an approximation for a certain target function under a class of distinguishers (test functions), where distinguishers are combinations of simple rectangle-indicators. In our case these approximations can be learned by a simple iterative procedure, which yields a unified and simple proof, achieving for any graph with density d and any approximation parameter $\epsilon$ the partition size

• a tower of 2’s of height $O\left( d_{}\epsilon ^{-2} \right)$ for a variant of Strong Regularity

• a power of 2 with exponent $O\left( d\epsilon ^{-2} \right)$ for Weak Regularity

The novelty in our proof is: (a) a simple approach which yields both strong and weaker variant, and (b) improvements when $d=o(1)$. At an abstract level, our proof can be seen a refinement and simplification of the “analytic” proof given by Lovasz and Szegedy.

This paper (with updates) is available on https://eprint.iacr.org/2016/965.pdf

M. Skórski—Supported by the European Research Council Consolidator Grant (682815-TOCNe).

A Proof of Lemma 2

Proof

Let d be the edge density of the pair (T, S) and $d'$ be the edge density of the pair $(T',S')$. Denote $\epsilon = \mathrm {irreg}_{G}(T,S)$. For any two subsets $T'' \subset T', S''\subset S'$, which are also subsets of T and S respectively, by the definition of d we have

$$\begin{aligned} \left| \frac{E(T',S')}{|T'||S'|} - d\right| \leqslant \epsilon . \end{aligned}$$

which translates to

$$\begin{aligned} |d'-d| \leqslant \epsilon . \end{aligned}$$

(19)

Therefore, by Eq. (19) and the triangle inequality

$$\begin{aligned} \left| E(T'',S'') - d'\cdot |T''||S''|\right|&\leqslant \left| E(T'',S'') - d\cdot |T''||S''|\right| + \epsilon \cdot |T''||S''|. \end{aligned}$$

(20)

Since the definition of d applied to $T''\subset T, S''\subset S$ implies

$$\begin{aligned}\left| E(T'',S'') - d'\cdot |T''||S''|\right| \leqslant \epsilon \cdot |T''||S''|, \end{aligned}$$

from Eq. (20) we conclude that

$$\begin{aligned} \left| E(T'',S'') - d'\cdot |T''||S''|\right|&\leqslant 2 \epsilon \cdot |T''||S''|, \end{aligned}$$

which finishes the proof.