DROPS

Document

DOI: 10.4230/LIPIcs.WABI.2026.19

On the Complexity of the (𝓁, k)-Median Problems

Authors: Luís Cunha, Thiago Nascimento, Marilia D. V. Braga, and Jens Stoye

Published in: LIPIcs, Volume 390, 26th International Conference on Algorithms for Bioinformatics (WABI 2026)

Abstract

The genome median problem is a central computational problem in comparative genomics, as it models the reconstruction of an ancestral genome from a set of related genomes. Given 𝓁 genomes and a distance measure, the problem asks for a genome that minimizes the sum of the distances to the input genomes. Two classical distances are the breakpoint distance and the double-cut-and-join (DCJ) distance. For multichromosomal circular genomes, the median problem is polynomial-time solvable under the breakpoint distance, whereas it is NP-hard under the DCJ distance. For even integer k ≥ 2, the σ_k distance interpolates between these two extremes: σ₂ corresponds to the breakpoint distance, while σ_∞ corresponds to the DCJ distance. A central open problem in this setting is the (3,4)-Median problem, which asks for a median of three genomes under the σ₄ distance, the first intermediate distance after the breakpoint distance. Motivated by this question, we study the more general (𝓁,k)-Median problem, in which 𝓁 is the number of input genomes and k determines the σ_k distance. We prove that (𝓁,6)-Median for every 𝓁 ≥ 4 and (3,12)-Median are NP-complete. We then extend the hardness of (𝓁,6)-Median to (𝓁,k)-Median for all even k ≥ 6, and the hardness of (3,12)-Median to (3,k)-Median for all even k ≥ 12. These results identify broad hardness regions in the (𝓁,k) parameter space and delimit the remaining open cases around the fundamental (3,4)-Median problem.

Cite as

Luís Cunha, Thiago Nascimento, Marilia D. V. Braga, and Jens Stoye. On the Complexity of the (𝓁, k)-Median Problems. In 26th International Conference on Algorithms for Bioinformatics (WABI 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 390, pp. 19:1-19:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

Copy BibTex To Clipboard

@InProceedings{cunha_et_al:LIPIcs.WABI.2026.19,
  author =	{Cunha, Lu{\'\i}s and Nascimento, Thiago and Braga, Marilia D. V. and Stoye, Jens},
  title =	{{On the Complexity of the (𝓁, k)-Median Problems}},
  booktitle =	{26th International Conference on Algorithms for Bioinformatics (WABI 2026)},
  pages =	{19:1--19:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-446-8},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{390},
  editor =	{El-Mabrouk, Nadia and Vandin, Fabio},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.WABI.2026.19},
  URN =		{urn:nbn:de:0030-drops-275231},
  doi =		{10.4230/LIPIcs.WABI.2026.19},
  annote =	{Keywords: Genome rearrangement, median problem, sigma-k distance}
}

Document

DOI: 10.4230/LIPIcs.WABI.2024.2

On the Complexity of the Median and Closest Permutation Problems

Authors: Luís Cunha, Ignasi Sau, and Uéverton Souza

Published in: LIPIcs, Volume 312, 24th International Workshop on Algorithms in Bioinformatics (WABI 2024)

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.

Cite as

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)

Copy BibTex To Clipboard

@InProceedings{cunha_et_al:LIPIcs.WABI.2024.2,
  author =	{Cunha, Lu{\'\i}s and Sau, Ignasi and Souza, U\'{e}verton},
  title =	{{On the Complexity of the Median and Closest Permutation Problems}},
  booktitle =	{24th International Workshop on Algorithms in Bioinformatics (WABI 2024)},
  pages =	{2:1--2:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-340-9},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{312},
  editor =	{Pissis, Solon P. and Sung, Wing-Kin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.WABI.2024.2},
  URN =		{urn:nbn:de:0030-drops-206468},
  doi =		{10.4230/LIPIcs.WABI.2024.2},
  annote =	{Keywords: Median problem, Closest problem, Genome rearrangements, Parameterized complexity}
}

Document

DOI: 10.4230/LIPIcs.CPM.2017.19

Fast and Simple Jumbled Indexing for Binary Run-Length Encoded Strings

Authors: Luís Cunha, Simone Dantas, Travis Gagie, Roland Wittler, Luis Kowada, and Jens Stoye

Published in: LIPIcs, Volume 78, 28th Annual Symposium on Combinatorial Pattern Matching (CPM 2017)

Abstract

Important papers have appeared recently on the problem of indexing binary strings for jumbled pattern matching, and further lowering the time bounds in terms of the input size would now be a breakthrough with broad implications. We can still make progress on the problem, however, by considering other natural parameters. Badkobeh et al. (IPL, 2013) and Amir et al. (TCS, 2016) gave algorithms that index a binary string in O(n + r^2 log r) time, where n is the length and r is the number of runs, and Giaquinta and Grabowski (IPL, 2013) gave one that runs in O(n + r^2) time. In this paper we propose a new and very simple algorithm that also runs in O(n + r^2) time and can be extended either so that the index returns the position of a match (if there is one), or so that the algorithm uses only O(n) bits of space instead of O(n) words.

Cite as

Luís Cunha, Simone Dantas, Travis Gagie, Roland Wittler, Luis Kowada, and Jens Stoye. Fast and Simple Jumbled Indexing for Binary Run-Length Encoded Strings. In 28th Annual Symposium on Combinatorial Pattern Matching (CPM 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 78, pp. 19:1-19:9, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

Copy BibTex To Clipboard

@InProceedings{cunha_et_al:LIPIcs.CPM.2017.19,
  author =	{Cunha, Lu{\'\i}s and Dantas, Simone and Gagie, Travis and Wittler, Roland and Kowada, Luis and Stoye, Jens},
  title =	{{Fast and Simple Jumbled Indexing for Binary Run-Length Encoded Strings}},
  booktitle =	{28th Annual Symposium on Combinatorial Pattern Matching (CPM 2017)},
  pages =	{19:1--19:9},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-039-2},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{78},
  editor =	{K\"{a}rkk\"{a}inen, Juha and Radoszewski, Jakub and Rytter, Wojciech},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CPM.2017.19},
  URN =		{urn:nbn:de:0030-drops-73418},
  doi =		{10.4230/LIPIcs.CPM.2017.19},
  annote =	{Keywords: string algorithms, indexing, jumbled pattern matching, run-length encoding}
}

Search Results

Documents authored by Cunha, Luís

On the Complexity of the (𝓁, k)-Median Problems

Abstract

Cite as

On the Complexity of the Median and Closest Permutation Problems

Abstract

Cite as

Fast and Simple Jumbled Indexing for Binary Run-Length Encoded Strings

Abstract

Cite as

Thanks for your feedback!

Could not send message