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On the Complexity of the Median and Closest Permutation Problems

Authors Luís Cunha , Ignasi Sau , Uéverton Souza

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Author Details

Luís Cunha
  • Instituto de Computação, Universidade Federal Fluminense, Brasil
Ignasi Sau
  • LIRMM, Université de Montpellier, CNRS, France
Uéverton Souza
  • Instituto de Computação, Universidade Federal Fluminense, Brasil

Funding

  • Cunha, Luís: FAPERJ-JCNE (E-26/201.372/2022), and CNPq-Universal (406173/2021-4).
  • Sau, Ignasi: project ELIT (ANR-20-CE48-0008-01), and CAPES/PRINT Programa Institucional de Internacionalização, edital nº 41/2017, grant 88887.717401/2022-00.
  • Souza, Uéverton: FAPERJ-JCNE (E-26/201.344/2021), and CNPq (309832/2020-9).

Cite AsGet BibTex

Luís Cunha, Ignasi Sau, and Uéverton Souza. On the Complexity of the Median and Closest Permutation Problems. In 24th International Workshop on Algorithms in Bioinformatics (WABI 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 312, pp. 2:1-2:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.WABI.2024.2

Abstract

Genome rearrangements are events where large blocks of DNA exchange places during evolution. The analysis of these events is a promising tool for understanding evolutionary genomics, providing data for phylogenetic reconstruction based on genome rearrangement measures. Many pairwise rearrangement distances have been proposed, based on finding the minimum number of rearrangement events to transform one genome into the other, using some predefined operation. When more than two genomes are considered, we have the more challenging problem of rearrangement-based phylogeny reconstruction. Given a set of genomes and a distance notion, there are at least two natural ways to define the "target" genome. On the one hand, finding a genome that minimizes the sum of the distances from this to any other, called the median genome. On the other hand, finding a genome that minimizes the maximum distance to any other, called the closest genome. Considering genomes as permutations of distinct integers, some distance metrics have been extensively studied. We investigate the median and closest problems on permutations over the following metrics: breakpoint distance, swap distance, block-interchange distance, short-block-move distance, and transposition distance. In biological applications some values are usually very small, such as the solution value d or the number k of input permutations. For each of these metrics and parameters d or k, we analyze the closest and the median problems from the viewpoint of parameterized complexity. We obtain the following results: NP-hardness for finding the median/closest permutation regarding some metrics of distance, even for only k = 3 permutations; Polynomial kernels for the problems of finding the median permutation of all studied metrics, considering the target distance d as parameter; NP-hardness result for finding the closest permutation by short-block-moves; FPT algorithms and infeasibility of polynomial kernels for finding the closest permutation for some metrics when parameterized by the target distance d.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Combinatorics
  • Theory of computation → Parameterized complexity and exact algorithms
  • Mathematics of computing → Permutations and combinations
Keywords
  • Median problem
  • Closest problem
  • Genome rearrangements
  • Parameterized complexity

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