Counting and Sampling Labeled Chordal Graphs in Polynomial Time
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(n⁷) 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.
Counting algorithms
graph sampling
chordal graphs
Theory of computation~Generating random combinatorial structures
Theory of computation~Graph algorithms analysis
58:1-58:17
Regular Paper
https://github.com/uhebertj/chordal
https://arxiv.org/abs/2308.09703
Úrsula
Hébert-Johnson
Úrsula Hébert-Johnson
University of California, Santa Barbara, CA, USA
https://orcid.org/0000-0001-8615-1253
Supported by NSF grant CCF-2008838.
Daniel
Lokshtanov
Daniel Lokshtanov
University of California, Santa Barbara, CA, USA
Supported by NSF grant CCF-2008838.
Eric
Vigoda
Eric Vigoda
University of California, Santa Barbara, CA, USA
Supported by NSF grant CCF-2147094.
10.4230/LIPIcs.ESA.2023.58
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Úrsula Hébert-Johnson, Daniel Lokshtanov, and Eric Vigoda
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