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**Published in:** LIPIcs, Volume 214, 16th International Symposium on Parameterized and Exact Computation (IPEC 2021)

We study the counting problem known as #PPM, whose input is a pair of permutations π and τ (called pattern and text, respectively), and the task is to find the number of subsequences of τ that have the same relative order as π. A simple brute-force approach solves #PPM for a pattern of length k and a text of length n in time O(n^{k+1}), while Berendsohn, Kozma and Marx have recently shown that under the exponential time hypothesis (ETH), it cannot be solved in time f(k) n^{o(k/log k)} for any function f. In this paper, we consider the restriction of #PPM, known as 𝒞-Pattern #PPM, where the pattern π must belong to a hereditary permutation class 𝒞. Our goal is to identify the structural properties of 𝒞 that determine the complexity of 𝒞-Pattern #PPM.
We focus on two such structural properties, known as the long path property (LPP) and the deep tree property (DTP). Assuming ETH, we obtain these results:
1) If 𝒞 has the LPP, then 𝒞-Pattern #PPM cannot be solved in time f(k)n^{o(√k)} for any function f, and
2) if 𝒞 has the DTP, then 𝒞-Pattern #PPM cannot be solved in time f(k)n^{o(k/log² k)} for any function f.
Furthermore, when 𝒞 is one of the so-called monotone grid classes, we show that if 𝒞 has the LPP but not the DTP, then 𝒞-Pattern #PPM can be solved in time f(k)n^{O(√ k)}. In particular, the lower bounds above are tight up to the polylog terms in the exponents.

Vít Jelínek, Michal Opler, and Jakub Pekárek. Long Paths Make Pattern-Counting Hard, and Deep Trees Make It Harder. In 16th International Symposium on Parameterized and Exact Computation (IPEC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 214, pp. 22:1-22:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{jelinek_et_al:LIPIcs.IPEC.2021.22, author = {Jel{\'\i}nek, V{\'\i}t and Opler, Michal and Pek\'{a}rek, Jakub}, title = {{Long Paths Make Pattern-Counting Hard, and Deep Trees Make It Harder}}, booktitle = {16th International Symposium on Parameterized and Exact Computation (IPEC 2021)}, pages = {22:1--22:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-216-7}, ISSN = {1868-8969}, year = {2021}, volume = {214}, editor = {Golovach, Petr A. and Zehavi, Meirav}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2021.22}, URN = {urn:nbn:de:0030-drops-154050}, doi = {10.4230/LIPIcs.IPEC.2021.22}, annote = {Keywords: Permutation pattern matching, subexponential algorithm, conditional lower bounds, tree-width} }

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**Published in:** LIPIcs, Volume 202, 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)

We study the complexity of the decision problem known as Permutation Pattern Matching, or PPM. The input of PPM consists of a pair of permutations τ (the "text") and π (the "pattern"), and the goal is to decide whether τ contains π as a subpermutation. On general inputs, PPM is known to be NP-complete by a result of Bose, Buss and Lubiw. In this paper, we focus on restricted instances of PPM where the text is assumed to avoid a fixed (small) pattern σ; this restriction is known as Av(σ)-PPM. It has been previously shown that Av(σ)-PPM is polynomial for any σ of size at most 3, while it is NP-hard for any σ containing a monotone subsequence of length four.
In this paper, we present a new hardness reduction which allows us to show, in a uniform way, that Av(σ)-PPM is hard for every σ of size at least 6, for every σ of size 5 except the symmetry class of 41352, as well as for every σ symmetric to one of the three permutations 4321, 4312 and 4231. Moreover, assuming the exponential time hypothesis, none of these hard cases of Av(σ)-PPM can be solved in time 2^o(n/log n). Previously, such conditional lower bound was not known even for the unconstrained PPM problem.
On the tractability side, we combine the CSP approach of Guillemot and Marx with the structural results of Huczynska and Vatter to show that for any monotone-griddable permutation class 𝒞, PPM is polynomial when the text is restricted to a permutation from 𝒞.

Vít Jelínek, Michal Opler, and Jakub Pekárek. Griddings of Permutations and Hardness of Pattern Matching. In 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 202, pp. 65:1-65:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{jelinek_et_al:LIPIcs.MFCS.2021.65, author = {Jel{\'\i}nek, V{\'\i}t and Opler, Michal and Pek\'{a}rek, Jakub}, title = {{Griddings of Permutations and Hardness of Pattern Matching}}, booktitle = {46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)}, pages = {65:1--65:22}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-201-3}, ISSN = {1868-8969}, year = {2021}, volume = {202}, editor = {Bonchi, Filippo and Puglisi, Simon J.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2021.65}, URN = {urn:nbn:de:0030-drops-145050}, doi = {10.4230/LIPIcs.MFCS.2021.65}, annote = {Keywords: Permutation, pattern matching, NP-hardness} }

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**Published in:** LIPIcs, Volume 170, 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)

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.

Vít Jelínek, Michal Opler, and Jakub Pekárek. A Complexity Dichotomy for Permutation Pattern Matching on Grid Classes. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 52:1-52:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{jelinek_et_al:LIPIcs.MFCS.2020.52, author = {Jel{\'\i}nek, V{\'\i}t and Opler, Michal and Pek\'{a}rek, Jakub}, 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 = {Esparza, Javier and Kr\'{a}l', Daniel}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.52}, URN = {urn:nbn:de:0030-drops-127186}, doi = {10.4230/LIPIcs.MFCS.2020.52}, annote = {Keywords: permutations, pattern matching, grid classes} }

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**Published in:** LIPIcs, Volume 112, 26th Annual European Symposium on Algorithms (ESA 2018)

A permutation pi is a merge of a permutation sigma and a permutation tau, if we can color the elements of pi red and blue so that the red elements have the same relative order as sigma and the blue ones as tau. We consider, for fixed hereditary permutation classes C and D, the complexity of determining whether a given permutation pi is a merge of an element of C with an element of D.
We develop general algorithmic approaches for identifying polynomially tractable cases of merge recognition. Our tools include a version of nondeterministic logspace streaming recognizability of permutations, which we introduce, and a concept of bounded width decomposition, inspired by the work of Ahal and Rabinovich.
As a consequence of the general results, we can provide nontrivial examples of tractable permutation merges involving commonly studied permutation classes, such as the class of layered permutations, the class of separable permutations, or the class of permutations avoiding a decreasing sequence of a given length.
On the negative side, we obtain a general hardness result which implies, for example, that it is NP-complete to recognize the permutations that can be merged from two subpermutations avoiding the pattern 2413.

Vít Jelínek, Michal Opler, and Pavel Valtr. Generalized Coloring of Permutations. In 26th Annual European Symposium on Algorithms (ESA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 112, pp. 50:1-50:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{jelinek_et_al:LIPIcs.ESA.2018.50, author = {Jel{\'\i}nek, V{\'\i}t and Opler, Michal and Valtr, Pavel}, title = {{Generalized Coloring of Permutations}}, booktitle = {26th Annual European Symposium on Algorithms (ESA 2018)}, pages = {50:1--50:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-081-1}, ISSN = {1868-8969}, year = {2018}, volume = {112}, editor = {Azar, Yossi and Bast, Hannah and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2018.50}, URN = {urn:nbn:de:0030-drops-95137}, doi = {10.4230/LIPIcs.ESA.2018.50}, annote = {Keywords: Permutations, merge, generalized coloring} }

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