Sampling a random permutation with restricted positions, or equivalently approximating the permanent of a 0-1 matrix, is a fundamental problem in computer science, with several notable results attained through the years. In this paper, we first improves the running time of the algorithms for a single permutation. We propose a fast approximation algorithm for the permanent of $\gamma$-dense 0-1 matrix, with an expected running time of $\tilde{O}\left(n^{2+(1-\gamma)/(2\gamma - 1)}\right)$. Our result removes the $n^4$ term from the previous best runtime and provides an improvement for $\gamma \geq 0.6$. When $\gamma = o(1)$, our runtime is $\tilde{\Theta}(n^2)$, which is nearly optimal for this problem. The core of our proof is to demonstrate that the Sinkhorn algorithm, a fundamental tool in matrix scaling, can achieve maximum accuracy of $1/\text{poly}(n)$ for dense matrices in $O(\log n)$ iterations. We further introduce a general model called permutations with disjunctive constraints (PDC) for handling multiple constrained permutations. We propose a novel Markov chain-based algorithm for sampling nearly uniform solutions of PDC within a Lov${\'a}$sz Local Lemma (LLL)-like regime by a novel sampling framework called correlated factorization. For uniform PDC formulas, where all constraints are of the same length and all permutations are of equal size, our algorithm runs in nearly linear time with respect to the number of variables.

翻译：采样具有受限位置的随机排列，等价于近似0-1矩阵的积和式，是计算机科学中的一个基础性问题，多年来已取得若干显著成果。本文首先改进了单排列采样算法的运行时间。针对γ-稠密0-1矩阵的积和式，我们提出了一种快速近似算法，其期望运行时间为$\tilde{O}\left(n^{2+(1-\gamma)/(2\gamma - 1)}\right)$。该结果消除了先前最佳运行时间中的$n^4$项，并在$\gamma \geq 0.6$时实现了性能提升。当$\gamma = o(1)$时，我们的运行时间为$\tilde{\Theta}(n^2)$，这在该问题上近乎最优。证明的核心在于展示矩阵缩放的基本工具——Sinkhorn算法，对于稠密矩阵能够在$O(\log n)$次迭代内达到$1/\text{poly}(n)$的最大精度。我们进一步引入了一种称为带析取约束的排列（PDC）的通用模型，用于处理多个受约束的排列。通过一种称为相关分解的新型采样框架，我们提出了一种基于马尔可夫链的算法，用于在类似Lov${\'a}$sz局部引理（LLL）的范围内采样近乎均匀的PDC解。对于均匀PDC公式（所有约束长度相同且所有排列规模相等），我们的算法在变量数量上具有近乎线性的运行时间。