期望多项式时间内无标号弦图的采样
Sampling Unlabeled Chordal Graphs in Expected Polynomial Time
AI总结：
本研究设计出期望多项式时间内均匀随机采样n顶点无标号弦图的算法，同时提出带给定自同构的有标号弦图计数采样的FPT算法，还证明了相关自同构的概率上界。
摘要：
我们设计了一种算法，可在期望多项式时间内均匀随机生成一个n顶点无标号弦图。在此过程中，我们得到了以下两项结果：（1）一种用于计数和采样具有给定自同构π的有标号弦图的FPT（固定参数可解）算法，其参数为π的移动点数量；（2）证明了随机n顶点有标号弦图具有给定自同构π∈Sₙ的概率至多为1/2^(c·max{μ²,n})，其中μ为π的移动点数量，c为常数。我们的无标号弦图采样算法将上述FPT算法作为黑盒调用，参数μ可能取较大值，但调用大μ值该算法的概率呈指数级小。
英文摘要：
We design an algorithm that generates an $n$-vertex unlabeled chordal graph uniformly at random in expected polynomial time. Along the way, we develop the following two results: (1) an $\mathsf{FPT}$ algorithm for counting and sampling labeled chordal graphs with a given automorphism $π$, parameterized by the number of moved points of $π$, and (2) a proof that the probability that a random $n$-vertex labeled chordal graph has a given automorphism $π\in S_n$ is at most $1/2^{c\max\{μ^2,n\}}$, where $μ$ is the number of moved points of $π$ and $c$ is a constant. Our algorithm for sampling unlabeled chordal graphs calls the aforementioned $\mathsf{FPT}$ algorithm as a black box with potentially large values of the parameter $μ$, but the probability of calling this algorithm with a large value of $μ$ is exponentially small.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
