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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, Alexander Wiedemann

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  • Theory of computation → Complexity classes
  • Mathematics of computing → Graph theory
Keywords
  • Genome Rearrangement
  • Phylogenetics
  • Single Cut-and-Join
  • Computational Complexity

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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) https://doi.org/10.4230/LIPIcs.SWAT.2024.3

Author Details

Lora Bailey
  • Department of Mathematics, Grand Valley State University, Allendale, MI, USA
Heather Smith Blake
  • Department of Mathematics and Computer Science, Davidson College, NC, USA
Garner Cochran
  • Department of Mathematics and Computer Science, Berry College, Mount Berry, GA, USA
Nathan Fox
  • Department of Quantitative Sciences, Canisius University, Buffalo, NY, USA
Michael Levet
  • Department of Computer Science, College of Charleston, SC, USA
Reem Mahmoud
  • Department of Computer Science, Virginia Commonwealth University, Richmond, VA, USA
Inne Singgih
  • Department of Mathematical Sciences, University of Cincinnati, OH, USA
Grace Stadnyk
  • Department of Mathematics, Furman University, Greenville, SC, USA
Alexander Wiedemann
  • Department of Mathematics, Randolph-Macon College, Ashland, VA, USA

Acknowledgements

We wish to thank the American Mathematical Society for organizing the Mathematics Research Community workshop where this work began. This material is based upon work supported by the National Science Foundation under Grant Number DMS 1641020.

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