Counting and Sampling Labeled Chordal Graphs in Polynomial Time
Counting and Sampling Labeled Chordal Graphs in Polynomial Time
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DOI：
10.48550/arxiv.2308.09703
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发表时间：
2023-08
期刊：
影响因子：
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通讯作者：
Úrsula Hébert-Johnson;D. Lokshtanov;Eric Vigoda
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文献类型：
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作者：
Úrsula Hébert-Johnson;D. Lokshtanov;Eric Vigoda
我们提出了第一个多项式时间算法来精确计算n个顶点上标记弦图的数量。我们的算法解决了一个更普遍的问题：给定n和作为输入，它使用O（n^7）个算术运算，计算n个顶点上可着色的标记弦图的数量。然后，一个标准的抽样到计数约简产生一个多项式时间的精确采样器，该采样器均匀随机地在n个顶点上生成一个可着色的标记弦图。我们的计数算法改进了Wormald（1985）之前的最佳结果，后者以n的时间指数计算n个顶点上标记的弦图的数量。多项式时间计数算法的实现在不到三分钟的时间内在标准台式计算机上给出了多达30个顶点上标记的弦图的数量。在此之前，标记弦图的数量只对最多15个顶点的图已知。
We present the first polynomial-time algorithm to exactly compute the number of labeled chordal graphs on n vertices. Our algorithm solves a more general problem: given n and omega as input, it computes the number of omega-colorable labeled chordal graphs on n vertices, using O(n^7) arithmetic operations. A standard sampling-to-counting reduction then yields a polynomial-time exact sampler that generates an omega-colorable labeled chordal graph on n vertices uniformly at random. Our counting algorithm improves upon the previous best result by Wormald (1985), which computes the number of labeled chordal graphs on n vertices in time exponential in n. An implementation of the polynomial-time counting algorithm gives the number of labeled chordal graphs on up to 30 vertices in less than three minutes on a standard desktop computer. Previously, the number of labeled chordal graphs was only known for graphs on up to 15 vertices.