License: 
Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ESA.2023.58
URN: urn:nbn:de:0030-drops-187119
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2023/18711/
Hébert-Johnson, Úrsula ;
Lokshtanov, Daniel ;
Vigoda, Eric
Counting and Sampling Labeled Chordal Graphs in Polynomial Time
Abstract
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.
BibTeX - Entry
@InProceedings{hebertjohnson_et_al:LIPIcs.ESA.2023.58,
author = {H\'{e}bert-Johnson, \'{U}rsula and Lokshtanov, Daniel and Vigoda, Eric},
title = {{Counting and Sampling Labeled Chordal Graphs in Polynomial Time}},
booktitle = {31st Annual European Symposium on Algorithms (ESA 2023)},
pages = {58:1--58:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-295-2},
ISSN = {1868-8969},
year = {2023},
volume = {274},
editor = {G{\o}rtz, Inge Li and Farach-Colton, Martin and Puglisi, Simon J. and Herman, Grzegorz},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2023/18711},
URN = {urn:nbn:de:0030-drops-187119},
doi = {10.4230/LIPIcs.ESA.2023.58},
annote = {Keywords: Counting algorithms, graph sampling, chordal graphs}
}
