eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2023-08-30
58:1
58:17
10.4230/LIPIcs.ESA.2023.58
article
Counting and Sampling Labeled Chordal Graphs in Polynomial Time
Hébert-Johnson, Úrsula
1
https://orcid.org/0000-0001-8615-1253
Lokshtanov, Daniel
1
Vigoda, Eric
1
University of California, Santa Barbara, CA, USA
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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol274-esa2023/LIPIcs.ESA.2023.58/LIPIcs.ESA.2023.58.pdf
Counting algorithms
graph sampling
chordal graphs