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| Main Authors: | , , |
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| Format: | Preprint |
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2023
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| Online Access: | https://arxiv.org/abs/2308.09703 |
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| _version_ | 1866910629767413760 |
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| author | Hebert-Johnson, Ursula Lokshtanov, Daniel Vigoda, Eric |
| author_facet | Hebert-Johnson, Ursula Lokshtanov, Daniel Vigoda, Eric |
| contents | 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 $ω$ as input, it computes the number of $ω$-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 $ω$-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.
In addition, we design two approximation algorithms: (1) an approximate counting algorithm that computes a $(1\pm\varepsilon)$-approximation of the number of $n$-vertex labeled chordal graphs, and (2) an approximate sampling algorithm that generates a random labeled chordal graph according to a distribution whose total variation distance from the uniform distribution is at most $\varepsilon$. The approximate counting algorithm runs in $O(n^3\log{n}\log^7(1/\varepsilon))$ time, and the approximate sampling algorithm runs in $O(n^3\log{n}\log^7(1/\varepsilon))$ expected time. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_09703 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Counting and Sampling Labeled Chordal Graphs in Polynomial Time Hebert-Johnson, Ursula Lokshtanov, Daniel Vigoda, Eric Data Structures and Algorithms 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 $ω$ as input, it computes the number of $ω$-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 $ω$-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. In addition, we design two approximation algorithms: (1) an approximate counting algorithm that computes a $(1\pm\varepsilon)$-approximation of the number of $n$-vertex labeled chordal graphs, and (2) an approximate sampling algorithm that generates a random labeled chordal graph according to a distribution whose total variation distance from the uniform distribution is at most $\varepsilon$. The approximate counting algorithm runs in $O(n^3\log{n}\log^7(1/\varepsilon))$ time, and the approximate sampling algorithm runs in $O(n^3\log{n}\log^7(1/\varepsilon))$ expected time. |
| title | Counting and Sampling Labeled Chordal Graphs in Polynomial Time |
| topic | Data Structures and Algorithms |
| url | https://arxiv.org/abs/2308.09703 |