On the sampling Lov\'asz Local Lemma for atomic constraint satisfaction problems

TL;DR

This paper introduces a nearly-linear time algorithm for sampling satisfying assignments in atomic constraint satisfaction problems under less restrictive conditions than previous methods, improving bounds for hypergraph colorings, Boolean formulas, and general CSPs.

Contribution

It presents a novel, less restrictive Lovász local lemma regime and a new information-percolation analysis for rapid mixing of Glauber dynamics in sampling.

Findings

Improved bounds for hypergraph q-colorings with /4 exponent.

Enhanced sampling bounds for Boolean k-CNF formulas.

New constant in the atomic CSP sampling regime.

Abstract

We study the problem of sampling an approximately uniformly random satisfying assignment for atomic constraint satisfaction problems i.e. where each constraint is violated by only one assignment to its variables. Let $p$ denote the maximum probability of violation of any constraint and let $\Delta$ denote the maximum degree of the line graph of the constraints. Our main result is a nearly-linear (in the number of variables) time algorithm for this problem, which is valid in a Lov\'asz local lemma type regime that is considerably less restrictive compared to previous works. In particular, we provide sampling algorithms for the uniform distribution on: (1) $q$-colorings of $k$-uniform hypergraphs with $\Delta \lesssim q^{(k-4)/3 + o_{q}(1)}.$ The exponent $1/3$ improves the previously best-known $1/7$ in the case $q, \Delta = O(1)$ [Jain, Pham, Vuong; arXiv, 2020] and $1/9$ in the general case [Feng, He, Yin; STOC 2021]. (2) Satisfying assignments of Boolean $k$-CNF formulas with $\Delta \lesssim 2^{k/5.741}.$ The constant $5.741$ in the exponent improves the previously best-known $7$ in the case $k = O(1)$ [Jain, Pham, Vuong; arXiv, 2020] and $13$ in the general case [Feng, He, Yin; STOC 2021]. (3) Satisfying assignments of general atomic constraint satisfaction problems with $p\cdot \Delta^{7.043} \lesssim 1.$ The constant $7.043$ improves upon the previously best-known constant of $350$ [Feng, He, Yin; STOC 2021]. At the heart of our analysis is a novel information-percolation type argument for showing the rapid mixing of the Glauber dynamics for a carefully constructed projection of the uniform distribution on satisfying assignments. Notably, there is no natural partial order on the space, and we believe that the techniques developed for the analysis may be of independent interest.