A Counting Lov\'asz Local Lemma

Hongyang Liu; Chunyang Wang; Yitong Yin; Yiyao Zhang; Can Zhou

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arXiv · 2608.08616

A Counting Lov\'asz Local Lemma

Hongyang Liu·Chunyang Wang·Yitong Yin·Yiyao Zhang·Can Zhou

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Abstract

We establish a counting analogue of the Lov\'asz Local Lemma: we give polynomial-time algorithms for approximately counting satisfying assignments of general constraint satisfaction problems (CSPs) in the local lemma regime $$ 4 \mathrm{e}\cdot p\cdot (D+1)^2\leq 1, $$ where $p$ is the maximum constraint violation probability and $D$ is the maximum dependency degree. This condition is tight up to constant factors, matching known lower bounds $pD^2\gtrsim 1$ for approximate counting in natural subclasses of CSPs. The core of our approach is a novel $2$-tree expansion for constraint marginal probabilities, which captures the decay of correlations in the local lemma regime.

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Hongyang Liu, Chunyang Wang, Yitong Yin, Yiyao Zhang, Can Zhou. 2026-08-09. A Counting Lov\'asz Local Lemma. https://arxiv.org/abs/2608.08616

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