Problem
Research questions and friction points this paper is trying to address.
constraint satisfaction problem
Lovász local lemma
FPRAS
sampling
A general counting and sampling Lovász local lemma
🤖 AI Summary
该研究解决了约束满足问题的概率估计与采样问题，通过扩展Lovász局部引理的方法，提出了一个在多项式期望时间内运行的近似采样器。
📝 Abstract
Consider a constraint satisfaction problem $\mathbf{C}$ on finitely many independent random variables with dependency graph $G$. Let $p_a$ be the violation probability of a constraint $a\in \mathbf{C}$ and $N_G^2 (a)$ the set of constraints at distance one or two from $a$ in $G$. Suppose that, there exists $x\in (0,1)^{\mathbf{C}}$ such that, for a sufficiently small universal constant $c > 0$, and for all $a \in \mathbf{C}$, \[ p_a \leq c \cdot x_a \prod_{b\in N_G^2(a)}(1-x_b). \] Under the above analog of the asymmetric Lovász Local Lemma, we give an FPRAS for the probability that all constraints are satisfied, and an approximate sampler, running in polynomial expected time, for the product distribution conditioned on this event. The degree of the polynomial in the running time is independent of the domain sizes, constraint sizes, or degree of the dependency graph. Up to the choice of the constant $c$, our condition on $p_a$ matches known hardness results.
Our work builds on the method of Liu, Wang, Yin, Zhang, and Zhou, who obtained an FPRAS for the probability of satisfaction in the setting of the symmetric Lovász Local Lemma. Our sampling result is new even in this special case.
Innovation
Methods, ideas, or system contributions that make the work stand out.
FPRAS
sampling
Lovász Local Lemma
constraint satisfaction problem
polynomial expected time
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