Sampling Unlabeled Chordal Graphs in Expected Polynomial Time
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866915101963976704 |
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| author | Hébert-Johnson, Úrsula Lokshtanov, Daniel |
| author_facet | Hébert-Johnson, Úrsula Lokshtanov, Daniel |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_05024 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Sampling Unlabeled Chordal Graphs in Expected Polynomial Time Hébert-Johnson, Úrsula Lokshtanov, Daniel Data Structures and Algorithms 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. |
| title | Sampling Unlabeled Chordal Graphs in Expected Polynomial Time |
| topic | Data Structures and Algorithms |
| url | https://arxiv.org/abs/2501.05024 |