License: Creative Commons Attribution 3.0 Unported license (CC-BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.MFCS.2020.52
URN: urn:nbn:de:0030-drops-127186
URL: https://drops.dagstuhl.de/opus/volltexte/2020/12718/
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Jelínek, Vít ; Opler, Michal ; Pekárek, Jakub

A Complexity Dichotomy for Permutation Pattern Matching on Grid Classes

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LIPIcs-MFCS-2020-52.pdf (0.7 MB)


Abstract

Permutation Pattern Matching (PPM) is the problem of deciding for a given pair of permutations π and τ whether the pattern π is contained in the text τ. Bose, Buss and Lubiw showed that PPM is NP-complete. In view of this result, it is natural to ask how the situation changes when we restrict the pattern π to a fixed permutation class 𝒞; this is known as the 𝒞-Pattern PPM problem. There have been several results in this direction, namely the work of Jelínek and Kynčl who completely resolved the hardness of 𝒞-Pattern PPM when 𝒞 is taken to be the class of σ-avoiding permutations for some σ. Grid classes are special kind of permutation classes, consisting of permutations admitting a grid-like decomposition into simpler building blocks. Of particular interest are the so-called monotone grid classes, in which each building block is a monotone sequence. Recently, it has been discovered that grid classes, especially the monotone ones, play a fundamental role in the understanding of the structure of general permutation classes. This motivates us to study the hardness of 𝒞-Pattern PPM for a (monotone) grid class 𝒞. We provide a complexity dichotomy for 𝒞-Pattern PPM when 𝒞 is taken to be a monotone grid class. Specifically, we show that the problem is polynomial-time solvable if a certain graph associated with 𝒞, called the cell graph, is a forest, and it is NP-complete otherwise. We further generalize our results to grid classes whose blocks belong to classes of bounded grid-width. We show that the 𝒞-Pattern PPM for such a grid class 𝒞 is polynomial-time solvable if the cell graph of 𝒞 avoids a cycle or a certain special type of path, and it is NP-complete otherwise.

BibTeX - Entry

@InProceedings{jelnek_et_al:LIPIcs:2020:12718,
  author =	{V{\'\i}t Jel{\'\i}nek and Michal Opler and Jakub Pek{\'a}rek},
  title =	{{A Complexity Dichotomy for Permutation Pattern Matching on Grid Classes}},
  booktitle =	{45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
  pages =	{52:1--52:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-159-7},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{170},
  editor =	{Javier Esparza and Daniel Kr{\'a}ľ},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2020/12718},
  URN =		{urn:nbn:de:0030-drops-127186},
  doi =		{10.4230/LIPIcs.MFCS.2020.52},
  annote =	{Keywords: permutations, pattern matching, grid classes}
}

Keywords: permutations, pattern matching, grid classes
Collection: 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)
Issue Date: 2020
Date of publication: 18.08.2020


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