A Sampling Lov\'{a}sz Local Lemma for Large Domain Sizes

TL;DR

This paper introduces polynomial-time algorithms for approximate counting and sampling in atomic CSPs with large domain sizes, nearly matching the theoretical lower bounds and advancing the understanding of the local lemma regime.

Contribution

It provides an almost tight sampling Lovász Local Lemma for large domain sizes, extending previous bounds and improving algorithmic capabilities for atomic CSPs.

Findings

Algorithms work in polynomial time for large domain sizes

Nearly matches the known lower bounds for approximate sampling

Establishes an almost tight local lemma regime for atomic CSPs

Abstract

We present polynomial-time algorithms for approximate counting and sampling solutions to constraint satisfaction problems (CSPs) with atomic constraints within the local lemma regime: $$ pD^{2+o_q(1)}\lesssim 1. $$ When the domain size $q$ of each variable becomes sufficiently large, this almost matches the known lower bound $pD^2\gtrsim 1$ for approximate counting and sampling solutions to atomic CSPs [Bez\'akov\'a et al, SICOMP '19; Galanis, Guo, Wang, TOCT '22], thus establishing an almost tight sampling Lov\'{a}sz local lemma for large domain sizes.