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Documents authored by Levet, Michael

Document
DOI: 10.4230/LIPIcs.SWAT.2026.30
Parallel Algorithms for Group Isomorphism via Code Equivalence

Authors: Michael Levet

Published in: LIPIcs, Volume 370, 20th Scandinavian Symposium on Algorithm Theory (SWAT 2026)

Abstract
In this paper, we exhibit AC³ isomorphism tests for coprime extensions H ⋉ N where H is elementary Abelian and N is Abelian; and groups where Rad(G) = Z(G) is elementary Abelian and G = Soc^{*}(G). The fact that isomorphism testing for these families is in P was established respectively by Qiao, Sarma, and Tang (STACS 2011), and Grochow and Qiao (CCC 2014, SIAM J. Comput. 2017). The polynomial-time isomorphism tests for both of these families crucially leveraged small (size O(log |G|)) instances of Linear Code Equivalence (Babai, SODA 2011). Here, we combine Luks' group-theoretic method for Graph Isomorphism (FOCS 1980, J. Comput. Syst. Sci. 1982) with the fact that G is given by its multiplication table, to implement the corresponding instances of Linear Code Equivalence in AC³. As a byproduct of our work, we show that isomorphism testing of arbitrary central-radical groups is decidable using AC circuits of depth O(log³ n) and size n^{O(log log n)}. This improves upon the previous bound of n^{O(log log n)}-time due to Grochow and Qiao (ibid.).
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Michael Levet. Parallel Algorithms for Group Isomorphism via Code Equivalence. In 20th Scandinavian Symposium on Algorithm Theory (SWAT 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 370, pp. 30:1-30:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{levet:LIPIcs.SWAT.2026.30,
  author =	{Levet, Michael},
  title =	{{Parallel Algorithms for Group Isomorphism via Code Equivalence}},
  booktitle =	{20th Scandinavian Symposium on Algorithm Theory (SWAT 2026)},
  pages =	{30:1--30:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-421-5},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{370},
  editor =	{Fraigniaud, Pierre},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2026.30},
  URN =		{urn:nbn:de:0030-drops-260660},
  doi =		{10.4230/LIPIcs.SWAT.2026.30},
  annote =	{Keywords: Group Isomorphism, Circuit Complexity, Code Equivalence}
}
Document
DOI: 10.4230/LIPIcs.SWAT.2024.3
Pairwise Rearrangement is Fixed-Parameter Tractable in the Single Cut-and-Join Model

Authors: Lora Bailey, Heather Smith Blake, Garner Cochran, Nathan Fox, Michael Levet, Reem Mahmoud, Inne Singgih, Grace Stadnyk, and Alexander Wiedemann

Published in: LIPIcs, Volume 294, 19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024)

Abstract
Genome rearrangement is a common model for molecular evolution. In this paper, we consider the Pairwise Rearrangement problem, which takes as input two genomes and asks for the number of minimum-length sequences of permissible operations transforming the first genome into the second. In the Single Cut-and-Join model (Bergeron, Medvedev, & Stoye, J. Comput. Biol. 2010), Pairwise Rearrangement is #P-complete (Bailey, et. al., COCOON 2023), which implies that exact sampling is intractable. In order to cope with this intractability, we investigate the parameterized complexity of this problem. We exhibit a fixed-parameter tractable algorithm with respect to the number of components in the adjacency graph that are not cycles of length 2 or paths of length 1. As a consequence, we obtain that Pairwise Rearrangement in the Single Cut-and-Join model is fixed-parameter tractable by distance. Our results suggest that the number of nontrivial components in the adjacency graph serves as the key obstacle for efficient sampling.
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Lora Bailey, Heather Smith Blake, Garner Cochran, Nathan Fox, Michael Levet, Reem Mahmoud, Inne Singgih, Grace Stadnyk, and Alexander Wiedemann. Pairwise Rearrangement is Fixed-Parameter Tractable in the Single Cut-and-Join Model. In 19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 294, pp. 3:1-3:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{bailey_et_al:LIPIcs.SWAT.2024.3,
  author =	{Bailey, Lora and Blake, Heather Smith and Cochran, Garner and Fox, Nathan and Levet, Michael and Mahmoud, Reem and Singgih, Inne and Stadnyk, Grace and Wiedemann, Alexander},
  title =	{{Pairwise Rearrangement is Fixed-Parameter Tractable in the Single Cut-and-Join Model}},
  booktitle =	{19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024)},
  pages =	{3:1--3:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-318-8},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{294},
  editor =	{Bodlaender, Hans L.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2024.3},
  URN =		{urn:nbn:de:0030-drops-200436},
  doi =		{10.4230/LIPIcs.SWAT.2024.3},
  annote =	{Keywords: Genome Rearrangement, Phylogenetics, Single Cut-and-Join, Computational Complexity}
}
Document
DOI: 10.4230/LIPIcs.SWAT.2024.32
Canonizing Graphs of Bounded Rank-Width in Parallel via Weisfeiler-Leman

Authors: Michael Levet, Puck Rombach, and Nicholas Sieger

Published in: LIPIcs, Volume 294, 19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024)

Abstract
In this paper, we show that computing canonical labelings of graphs of bounded rank-width is in TC². Our approach builds on the framework of Köbler & Verbitsky (CSR 2008), who established the analogous result for graphs of bounded treewidth. Here, we use the framework of Grohe & Neuen (ACM Trans. Comput. Log., 2023) to enumerate separators via split-pairs and flip functions. In order to control the depth of our circuit, we leverage the fact that any graph of rank-width k admits a rank decomposition of width ≤ 2k and height O(log n) (Courcelle & Kanté, WG 2007). This allows us to utilize an idea from Wagner (CSR 2011) of tracking the depth of the recursion in our computation. Furthermore, after splitting the graph into connected components, it is necessary to decide isomorphism of said components in TC¹. To this end, we extend the work of Grohe & Neuen (ibid.) to show that the (6k+3)-dimensional Weisfeiler-Leman (WL) algorithm can identify graphs of rank-width k using only O(log n) rounds. As a consequence, we obtain that graphs of bounded rank-width are identified by FO + C formulas with 6k+4 variables and quantifier depth O(log n). Prior to this paper, isomorphism testing for graphs of bounded rank-width was not known to be in NC.
Cite as

Michael Levet, Puck Rombach, and Nicholas Sieger. Canonizing Graphs of Bounded Rank-Width in Parallel via Weisfeiler-Leman. In 19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 294, pp. 32:1-32:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{levet_et_al:LIPIcs.SWAT.2024.32,
  author =	{Levet, Michael and Rombach, Puck and Sieger, Nicholas},
  title =	{{Canonizing Graphs of Bounded Rank-Width in Parallel via Weisfeiler-Leman}},
  booktitle =	{19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024)},
  pages =	{32:1--32:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-318-8},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{294},
  editor =	{Bodlaender, Hans L.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2024.32},
  URN =		{urn:nbn:de:0030-drops-200724},
  doi =		{10.4230/LIPIcs.SWAT.2024.32},
  annote =	{Keywords: Graph Isomorphism, Weisfeiler-Leman, Rank-Width, Canonization, Descriptive Complexity, Circuit Complexity}
}
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