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DOI: 10.4230/LIPIcs.STACS.2025.46

Sampling Unlabeled Chordal Graphs in Expected Polynomial Time

Authors: Úrsula Hébert-Johnson and Daniel Lokshtanov

Published in: LIPIcs, Volume 327, 42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025)

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)

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@InProceedings{hebertjohnson_et_al:LIPIcs.STACS.2025.46,
  author =	{H\'{e}bert-Johnson, \'{U}rsula and Lokshtanov, Daniel},
  title =	{{Sampling Unlabeled Chordal Graphs in Expected Polynomial Time}},
  booktitle =	{42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025)},
  pages =	{46:1--46:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-365-2},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{327},
  editor =	{Beyersdorff, Olaf and Pilipczuk, Micha{\l} and Pimentel, Elaine and Thắng, Nguy\~{ê}n Kim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2025.46},
  URN =		{urn:nbn:de:0030-drops-228726},
  doi =		{10.4230/LIPIcs.STACS.2025.46},
  annote =	{Keywords: Chordal graphs, graph sampling, graph counting, unlabeled graphs}
}

Document

DOI: 10.4230/LIPIcs.ESA.2023.58

Counting and Sampling Labeled Chordal Graphs in Polynomial Time

Authors: Úrsula Hébert-Johnson, Daniel Lokshtanov, and Eric Vigoda

Published in: LIPIcs, Volume 274, 31st Annual European Symposium on Algorithms (ESA 2023)

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.

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Úrsula Hébert-Johnson, Daniel Lokshtanov, and Eric Vigoda. Counting and Sampling Labeled Chordal Graphs in Polynomial Time. In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 58:1-58:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@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/entities/document/10.4230/LIPIcs.ESA.2023.58},
  URN =		{urn:nbn:de:0030-drops-187119},
  doi =		{10.4230/LIPIcs.ESA.2023.58},
  annote =	{Keywords: Counting algorithms, graph sampling, chordal graphs}
}

Document

DOI: 10.4230/LIPIcs.ISAAC.2021.32

Anonymity-Preserving Space Partitions

Authors: Úrsula Hébert-Johnson, Chinmay Sonar, Subhash Suri, and Vaishali Surianarayanan

Published in: LIPIcs, Volume 212, 32nd International Symposium on Algorithms and Computation (ISAAC 2021)

Abstract

We consider a multidimensional space partitioning problem, which we call Anonymity-Preserving Partition. Given a set P of n points in ℝ^d and a collection H of m axis-parallel hyperplanes, the hyperplanes of H partition the space into an arrangement A(H) of rectangular cells. Given an integer parameter t > 0, we call a cell C in this arrangement deficient if 0 < |C ∩ P| < t; that is, the cell contains at least one but fewer than t data points of P. Our problem is to remove the minimum number of hyperplanes from H so that there are no deficient cells. We show that the problem is NP-complete for all dimensions d ≥ 2. We present a polynomial-time d-approximation algorithm, for any fixed d, and we also show that the problem can be solved exactly in time (2d-0.924)^k m^O(1) + O(n), where k is the solution size. The one-dimensional case of the problem, where all hyperplanes are parallel, can be solved optimally in polynomial time, but we show that a related Interval Anonymity problem is NP-complete even in one dimension.

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Úrsula Hébert-Johnson, Chinmay Sonar, Subhash Suri, and Vaishali Surianarayanan. Anonymity-Preserving Space Partitions. In 32nd International Symposium on Algorithms and Computation (ISAAC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 212, pp. 32:1-32:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{hebertjohnson_et_al:LIPIcs.ISAAC.2021.32,
  author =	{H\'{e}bert-Johnson, \'{U}rsula and Sonar, Chinmay and Suri, Subhash and Surianarayanan, Vaishali},
  title =	{{Anonymity-Preserving Space Partitions}},
  booktitle =	{32nd International Symposium on Algorithms and Computation (ISAAC 2021)},
  pages =	{32:1--32:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-214-3},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{212},
  editor =	{Ahn, Hee-Kap and Sadakane, Kunihiko},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2021.32},
  URN =		{urn:nbn:de:0030-drops-154654},
  doi =		{10.4230/LIPIcs.ISAAC.2021.32},
  annote =	{Keywords: Anonymity, Hitting Set, LP, Constant Approximation, Fixed-Parameter Tractable, Space Partitions, Parameterized Complexity}
}

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Documents authored by Hébert-Johnson, Úrsula

Sampling Unlabeled Chordal Graphs in Expected Polynomial Time

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Counting and Sampling Labeled Chordal Graphs in Polynomial Time

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Anonymity-Preserving Space Partitions

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