Sampling and counting triangle-free graphs near the critical density

TL;DR

This paper investigates efficient algorithms for sampling and counting triangle-free graphs in the Erdős-Rényi model near the critical density, revealing structural impacts on algorithm performance in this regime.

Contribution

It introduces two novel algorithms tailored for different regimes of edge probability, advancing understanding of forbidden substructure sampling near the critical density.

Findings

Efficient sampling algorithms are developed for different regimes of p.

The algorithms perform well near the critical density p ~ n^{-1/2}.

Structural changes in G(n,p) influence sampling complexity.

Abstract

We study the following combinatorial counting and sampling problems: can we efficiently sample from the Erd\H{o}s-R\'{e}nyi random graph $G(n,p)$ conditioned on triangle-freeness? Can we efficiently approximate the probability that $G(n,p)$ is triangle-free? These are prototypical instances of forbidden substructure problems ubiquitous in combinatorics. The algorithmic questions are instances of approximate counting and sampling for a hypergraph hard-core model. Estimating the probability that $G(n,p)$ has no triangles is a fundamental question in probabilistic combinatorics and one that has led to the development of many important tools in the field. Through the work of several authors, the asymptotics of the logarithm of this probability are known if $p =o( n^{-1/2})$ or if $p =\omega( n^{-1/2})$. The regime $p = \Theta(n^{-1/2})$ is more mysterious, as this range witnesses a dramatic change in the the typical structural properties of $G(n,p)$ conditioned on triangle-freeness. As we show, this change in structure has a profound impact on the performance of sampling algorithms. We give two different efficient sampling algorithms for triangle-free graphs (and complementary algorithms to approximate the triangle-freeness large deviation probability), one that is efficient when $p < c/\sqrt{n}$ and one that is efficient when $p > C/\sqrt{n}$ for constants $c, C>0$. The latter algorithm involves a new approach for dealing with large defects in the setting of sampling from low-temperature spin models.