A Sampling Lovász Local Lemma for Large Domain Sizes
Chunyang Wang, Yitong Yin
TL;DR
This work advances counting and sampling for atomic CSPs under a local lemma regime by introducing a constraint-wise coupling that exhibits exponential decay of correlation as domain sizes grow, enabling a new LP-based framework to bootstrap marginal probabilities. The combination yields deterministic approximate counting and almost-uniform sampling in time $O\left((n/\varepsilon)^{\mathrm{poly}(k,D,\log q_{\max})}\right)$ under the regime $(8e)^{1+\zeta/2} p (D+1)^{2+\zeta} \le 1$, with $\zeta$ diminishing as minimum domain size increases, approaching the conjectured $pD^2 \lesssim 1$ threshold for large domains. The paper also derives concrete consequences for hypergraph q-colorings and k-CNF formulas, tightening the known bounds on the counting/sampling Lovász local lemma in these canonical atomic CSPs. A key innovation is replacing the traditional freezing/factorization approaches with a constraint-wise self-reducibility LP that leverages 2-tree overflow constraints, offering a new lens on the LLL for counting and sampling. Overall, the results push the algorithmic frontier closer to the information-theoretic limits and open avenues for extending to general CSPs and smaller domain sizes with further refinements.
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 [Bezáková et al, SICOMP '19; Galanis, Guo, Wang, TOCT '22], thus establishing an almost tight sampling Lovász local lemma for large domain sizes.