Sampling Colorings and Independent Sets of Random Regular Bipartite Graphs in the Non-Uniqueness Region

TL;DR

This paper develops new algorithms for approximating partition functions and sampling from Gibbs distributions for spin systems on random regular bipartite graphs in the non-uniqueness region, improving previous bounds significantly.

Contribution

It introduces an FPRAS for q-colorings and the hard-core model in the non-uniqueness region on random bipartite graphs, extending the applicability of polymer methods.

Findings

FPRAS for q-colorings with q=O(Δ/log Δ) on almost all Δ-regular bipartite graphs

FPRAS for the hard-core model with λ=Ω(log Δ)/Δ on random bipartite graphs

Improved bounds close to the best possible using polymer method techniques

Abstract

For spin systems, such as the $q$-colorings and independent-set models, approximating the partition function in the so-called non-uniqueness region, where the model exhibits long-range correlations, is typically computationally hard for bounded-degree graphs. We present new algorithmic results for approximating the partition function and sampling from the Gibbs distribution for spin systems in the non-uniqueness region on random regular bipartite graphs. We give an $\mathsf{FPRAS}$ for counting $q$-colorings for even $q=O\big(\tfrac{\Delta}{\log{\Delta}}\big)$ on almost every $\Delta$-regular bipartite graph. This is within a factor $O(\log{\Delta})$ of the sampling algorithm for general graphs in the uniqueness region and improves significantly upon the previous best bound of $q=O\big(\tfrac{\sqrt{\Delta}}{(\log\Delta)^2}\big)$ by Jenssen, Keevash, and Perkins (SODA'19). Analogously, for the hard-core model on independent sets weighted by $\lambda>0$, we present an $\mathsf{FPRAS}$ for estimating the partition function when $\lambda=\Omega\big(\tfrac{\log{\Delta}}{\Delta}\big)$, which improves upon previous results by an $\Omega(\log \Delta)$ factor. Our results for the colorings and hard-core models follow from a general result that applies to arbitrary spin systems. Our main contribution is to show how to elevate probabilistic/analytic bounds on the marginal probabilities for the typical structure of phases on random bipartite regular graphs into efficient algorithms, using the polymer method. We further show evidence that our result for colorings is within a constant factor of best possible using current polymer-method approaches.