在分级偏序集中近似采样具有固定秩的元素
Approximately Sampling Elements with Fixed Rank in Graded Posets
AI总结：
针对分级偏序集中通常为#P-完全的固定秩计数问题，提出基于balanced bias的带偏Markov chains近似均匀采样方法，在期望多项式时间内生成任意固定秩样本，并首次实现受限整数划分的高效均匀采样。
摘要：
分级偏序集在组合数学中频繁出现，很自然地会尝试计算固定秩元素的数量。这些计数问题通常是 $\\#\textbf{P}$-完全的，因此我们考虑用于计数和均匀采样的近似算法。我们证明，对于某些类别的偏序集，沿着其 Hasse 图的边游走的带偏 Markov chains 允许我们在期望多项式时间内近似生成具有任意固定秩的样本。我们的论证不依赖于对数凹性的典型证明，这些证明用于构造具有特定模式的平稳分布，以便为输出所需秩元素的概率提供下界。相反，我们通过一种称为 $\textit{balanced bias}$ 的方法，直接从链的混合时间界限中推断出这一点。我们方法的一个值得注意的应用是对 $n$ 的受限类整数划分进行采样。我们给出了第一个可证明高效的 Markov chain 算法，用于从一般受限类中均匀采样 $n$ 的整数划分。一些观察使我们能够提高该链的效率，使其需要 $O(n^{1/2}\log(n))$ 空间，而对于无限制整数划分，需要期望的 $O(n^{9/4})$ 时间。相关应用包括采样具有固定逆序数的排列，以及三角格上具有固定平均高度的菱形铺嵌。
英文摘要：
Graded posets frequently arise throughout combinatorics, where it is natural to try to count the number of elements of a fixed rank. These counting problems are often $\#\textbf{P}$-complete, so we consider approximation algorithms for counting and uniform sampling. We show that for certain classes of posets, biased Markov chains that walk along edges of their Hasse diagrams allow us to approximately generate samples with any fixed rank in expected polynomial time. Our arguments do not rely on the typical proofs of log-concavity, which are used to construct a stationary distribution with a specific mode in order to give a lower bound on the probability of outputting an element of the desired rank. Instead, we infer this directly from bounds on the mixing time of the chains through a method we call $\textit{balanced bias}$. A noteworthy application of our method is sampling restricted classes of integer partitions of $n$. We give the first provably efficient Markov chain algorithm to uniformly sample integer partitions of $n$ from general restricted classes. Several observations allow us to improve the efficiency of this chain to require $O(n^{1/2}\log(n))$ space, and for unrestricted integer partitions, expected $O(n^{9/4})$ time. Related applications include sampling permutations with a fixed number of inversions and lozenge tilings on the triangular lattice with a fixed average height.
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