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Parity Permutation Pattern Matching

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WALCOM: Algorithms and Computation (WALCOM 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13973))

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Abstract

Given two permutations, a pattern \(\sigma \) and a text \(\pi \), Parity Permutation Pattern Matching asks whether there exists a parity and order preserving embedding of \(\sigma \) into \(\pi \). While it is known that Permutation Pattern Matching is in \(\textsf{FPT}\), we show that adding the parity constraint to the problem makes it \(\textsf{W}[1]\)-hard, even for alternating permutations or for 4321-avoiding patterns. However, it remains in \(\textsf{FPT}\) if the text avoids a fixed permutation, thanks to a recent meta-theorem on twin-width. On the other hand, as for the classical version, Parity Permutation Pattern Matching remains polynomial-time solvable when both permutations are separable, or if both are 321-avoiding, but \(\textsf{NP}\)-hard if the pattern is 321-avoiding and the text is 4321-avoiding.

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References

  1. Ahal, S., Rabinovich, Y.: On complexity of the subpattern problem. SIAM J. Discrete Math. 22(2), 629–649 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  2. Albert, M.H., Lackner, M., Lackner, M., Vatter, V.: The complexity of pattern matching for 321-avoiding and skew-merged permutations. Discrete Math. Theor. Comput. Sci. 18(2) (2016)

    Google Scholar 

  3. Alexandersson, P., Fufa, S.A., Getachew, F., Qiu, D.: Pattern-avoidance and fuss-catalan numbers. arXiv preprint. arXiv:2201.08168 (2022)

  4. Berendsohn, B.A., Kozma, L., Marx, D.: Finding and counting permutations via CSPs. Algorithmica 83(8), 2552–2577 (2021). https://doi.org/10.1007/s00453-021-00812-z

    Article  MathSciNet  MATH  Google Scholar 

  5. Bonnet, É., Kim, E.J., Thomassé, S., Watrigant, R.: Twin-width I: tractable FO model checking. J. ACM 69(1), 3:1–3:46 (2022)

    Google Scholar 

  6. Bose, P., Buss, J.F., Lubiw, A.: Pattern matching for permutations. Inf. Process. Lett. 65(5), 277–283 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bruner, M.L., Lackner, M.: The computational landscape of permutation patterns. Pure Mathematics and Applications: Special Issue for the Permutation Patterns 2012 Conference, vol. 24, no. 2, pp. 83–101 (2013)

    Google Scholar 

  8. Bruner, M., Lackner, M.: A fast algorithm for permutation pattern matching based on alternating runs. Algorithmica 75(1), 84–117 (2016). https://doi.org/10.1007/s00453-015-0013-y

    Article  MathSciNet  MATH  Google Scholar 

  9. Bulteau, L., Fertin, G., Jugé, V., Vialette, S.: Permutation pattern matching for doubly partially ordered patterns. In: Proceedings of the CPM. LIPIcs, vol. 223, pp. 21:1–21:17 (2022)

    Google Scholar 

  10. Cormen, T.H., Leiserson, C.E., Rivest, R.L., Stein, C.: Introduction to Algorithms. MIT Press, Cambridge (2022)

    MATH  Google Scholar 

  11. Cygan, M., et al.: Parameterized Algorithms. Springer, Cham (2015)

    Book  MATH  Google Scholar 

  12. Fox, J.: Stanley-wilf limits are typically exponential. arXiv preprint. arXiv:1310.8378 (2013)

  13. Gawrychowski, P., Rzepecki, M.: Faster exponential algorithm for permutation pattern matching. In: 5th SOSA@SODA 2022, pp. 279–284. SIAM (2022)

    Google Scholar 

  14. Gil, J.B., Tomasko, J.A.: Restricted Grassmannian permutations. Enumerative Comb. Appl. 2(4), #S4PP6 (2021)

    Google Scholar 

  15. Guillemot, S., Marx, D.: Finding small patterns in permutations in linear time. In: Proceedings of the SODA, pp. 82–101. SIAM (2014)

    Google Scholar 

  16. Guillemot, S., Vialette, S.: Pattern matching for 321-avoiding permutations. In: Dong, Y., Du, D.-Z., Ibarra, O. (eds.) ISAAC 2009. LNCS, vol. 5878, pp. 1064–1073. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-10631-6_107

    Chapter  Google Scholar 

  17. Jelínek, V., Kynčl, J.: Hardness of permutation pattern matching. In: Proceedings of the SODA, pp. 378–396. SIAM (2017)

    Google Scholar 

  18. Jelínek, V., Opler, M., Pekárek, J.: Griddings of permutations and hardness of pattern matching. In: Proceedings of the MFCS. LIPIcs, vol. 202, pp. 65:1–65:22 (2021)

    Google Scholar 

  19. Kitaev, S.: Patterns in Permutations and Words. Monographs in Theoretical Computer Science. An EATCS Series, Springer, Berlin (2011). https://doi.org/10.1007/978-3-642-17333-2

    Book  MATH  Google Scholar 

  20. Tanimoto, S.: Combinatorics of the group of parity alternating permutations. Adv. Appl. Math. 44(3), 225–230 (2010)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

Thanks to Édouard Bonnet and Eun Jung Kim for pointing out the link with the twin-width framework, and to the reviewers for their useful comments.

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Authors and Affiliations

  1. Université Paris-Dauphine, PSL University, CNRS, LAMSADE, 75016, Paris, France

    Virginia Ardévol Martínez & Florian Sikora

  2. LIGM, CNRS, Univ Gustave Eiffel, 77454, Marne-la-Vallée, France

    Stéphane Vialette

Authors
  1. Virginia Ardévol Martínez
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  2. Florian Sikora
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  3. Stéphane Vialette
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Correspondence to Virginia Ardévol Martínez .

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Editors and Affiliations

  1. National Yang Ming Chiao Tung University, Hsinchu, Taiwan

    Chun-Cheng Lin

  2. National Yang Ming Chiao Tung University, Hsinchu, Taiwan

    Bertrand M. T. Lin

  3. University of Perugia, Perugia, Italy

    Giuseppe Liotta

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Ardévol Martínez, V., Sikora, F., Vialette, S. (2023). Parity Permutation Pattern Matching. In: Lin, CC., Lin, B.M.T., Liotta, G. (eds) WALCOM: Algorithms and Computation. WALCOM 2023. Lecture Notes in Computer Science, vol 13973. Springer, Cham. https://doi.org/10.1007/978-3-031-27051-2_32

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  • DOI: https://doi.org/10.1007/978-3-031-27051-2_32

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-27050-5

  • Online ISBN: 978-3-031-27051-2

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