发表机构
University of Illinois Chicago; University of Chicago（伊利诺伊大学芝加哥分校; 芝加哥大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对满足非对称Lovász局部引理条件的约束满足问题，提出计算满足概率的FPRAS和近似采样器，运行时间为多项式且与问题规模参数无关，采样结果在对称情形下也是新的。
AI 中文摘要
考虑一个在有限多个独立随机变量上的约束满足问题 $\mathbf{C}$，其依赖图为 $G$。设 $p_a$ 为约束 $a\in \mathbf{C}$ 的违反概率，$N_G^2 (a)$ 为 $G$ 中与 $a$ 距离为一或二的约束集合。假设存在 $x\in (0,1)^{\mathbf{C}}$，使得对于足够小的通用常数 $c > 0$，且对所有 $a \in \mathbf{C}$，有 \\[ p_a \leq c \cdot x_a \prod_{b\in N_G^2(a)}(1-x_b). \\] 在上述非对称Lovász局部引理的类比条件下，我们给出了所有约束都被满足的概率的FPRAS，以及一个在多项式期望时间内运行的近似采样器，用于在该事件条件下的乘积分布。运行时间中多项式的次数与定义域大小、约束大小或依赖图的度无关。除了常数 $c$ 的选择外，我们对 $p_a$ 的条件与已知的困难性结果相匹配。我们的工作基于Liu、Wang、Yin、Zhang和Zhou的方法，他们在对称Lovász局部引理的设置下获得了满足概率的FPRAS。我们的采样结果即使在这种特殊情况下也是新的。
英文摘要
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.
