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Sampling Unlabeled Chordal Graphs in Expected Polynomial Time

Authors Úrsula Hébert-Johnson , Daniel Lokshtanov

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ACM Subject Classification
  • Theory of computation → Generating random combinatorial structures
  • Theory of computation → Graph algorithms analysis
Keywords
  • Chordal graphs
  • graph sampling
  • graph counting
  • unlabeled graphs

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Abstract

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 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 ∈ S_n is at most 1/2^{c max{μ²,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 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.

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Úrsula Hébert-Johnson and Daniel Lokshtanov. Sampling Unlabeled Chordal Graphs in Expected Polynomial Time. In 42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 327, pp. 46:1-46:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.STACS.2025.46

Author Details

Úrsula Hébert-Johnson
  • University of California, Santa Barbara, CA, USA
Daniel Lokshtanov
  • University of California, Santa Barbara, CA, USA

Funding

  • Hébert-Johnson, Úrsula: Supported by NSF grant CCF-2147094.

Acknowledgements

We would like to thank Eric Vigoda for discussions that led to a concise proof of an important step in the running time analysis.

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