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A Complexity Dichotomy for Permutation Pattern Matching on Grid Classes

Authors Vít Jelínek , Michal Opler , Jakub Pekárek

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Author Details

Vít Jelínek
  • Computer Science Institute, Charles University, Prague, Czech Republic
Michal Opler
  • Computer Science Institute, Charles University, Prague, Czech Republic
Jakub Pekárek
  • Department of Applied Mathematics, Charles University, Prague, Czech Republic

Funding

Supported by the GAUK project 1766318.
  • Jelínek, Vít: Supported by project 18-19158S of the Czech Science Foundation.
  • Opler, Michal: Supported by project 18-19158S of the Czech Science Foundation.

Cite AsGet BibTex

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)
https://doi.org/10.4230/LIPIcs.MFCS.2020.52

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.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Permutations and combinations
  • Theory of computation → Pattern matching
  • Theory of computation → Problems, reductions and completeness
Keywords
  • permutations
  • pattern matching
  • grid classes

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References

  1. Shlomo Ahal and Yuri Rabinovich. On the complexity of the sub-permutation problem. In Algorithm theory - SWAT 2000 (Bergen), volume 1851 of Lecture Notes in Comput. Sci., pages 490-503. Springer, Berlin, 2000. URL: https://doi.org/10.1007/3-540-44985-X_41.
  2. Michael Albert, Jay Pantone, and Vincent Vatter. On the growth of merges and staircases of permutation classes. Rocky Mountain J. Math., 49(2):355-367, 2019. URL: https://doi.org/10.1216/RMJ-2019-49-2-355.
  3. Michael H. Albert, Robert E. L. Aldred, Mike D. Atkinson, and Derek A. Holton. Algorithms for pattern involvement in permutations. In Algorithms and computation (Christchurch, 2001), volume 2223 of Lecture Notes in Comput. Sci., pages 355-366. Springer, Berlin, 2001. URL: https://doi.org/10.1007/3-540-45678-3_31.
  4. Michael H. Albert, M. D. Atkinson, Mathilde Bouvel, Nik Ruškuc, and Vincent Vatter. Geometric grid classes of permutations. Trans. Amer. Math. Soc., 365(11):5859-5881, 2013. URL: https://doi.org/10.1090/S0002-9947-2013-05804-7.
  5. Michael H. Albert, Steve Linton, and Nik Ruškuc. The insertion encoding of permutations. Electron. J. Combin., 12:Research Paper 47, 31, 2005. URL: http://www.combinatorics.org/Volume_12/Abstracts/v12i1r47.html.
  6. M. D. Atkinson, M. M. Murphy, and N. Ruškuc. Partially well-ordered closed sets of permutations. Order, 19(2):101-113, 2002. URL: https://doi.org/10.1023/A:1016500300436.
  7. Christian Bean, Bjarki Gudmundsson, and Henning Ulfarsson. Automatic discovery of structural rules of permutation classes. Math. Comp., 88(318):1967-1990, 2019. URL: https://doi.org/10.1090/mcom/3386.
  8. David Bevan. Growth rates of geometric grid classes of permutations. Electron. J. Combin., 21(4):Paper 4.51, 17, 2014. Google Scholar
  9. David Bevan. Growth rates of permutation grid classes, tours on graphs, and the spectral radius. Trans. Amer. Math. Soc., 367(8):5863-5889, 2015. URL: https://doi.org/10.1090/S0002-9947-2015-06280-1.
  10. Prosenjit Bose, Jonathan F. Buss, and Anna Lubiw. Pattern matching for permutations. Inform. Process. Lett., 65(5):277-283, 1998. URL: https://doi.org/10.1016/S0020-0190(97)00209-3.
  11. Robert Brignall. Grid classes and partial well order. J. Combin. Theory Ser. A, 119(1):99-116, 2012. URL: https://doi.org/10.1016/j.jcta.2011.08.005.
  12. Marie-Louise Bruner and Martin Lackner. A fast algorithm for permutation pattern matching based on alternating runs. Algorithmica, 75(1):84-117, 2016. URL: https://doi.org/10.1007/s00453-015-0013-y.
  13. P. Erdös and G. Szekeres. A combinatorial problem in geometry. Compositio Math., 2:463-470, 1935. URL: http://www.numdam.org/item?id=CM_1935__2__463_0.
  14. Jacob Fox. Stanley-wilf limits are typically exponential. CoRR, abs/1310.8378, 2013. URL: http://arxiv.org/abs/1310.8378.
  15. Sylvain Guillemot and Dániel Marx. Finding small patterns in permutations in linear time. In Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 82-101. ACM, New York, 2014. URL: https://doi.org/10.1137/1.9781611973402.7.
  16. Sylvain Guillemot and Stéphane Vialette. Pattern matching for 321-avoiding permutations. In Algorithms and computation, volume 5878 of Lecture Notes in Comput. Sci., pages 1064-1073. Springer, Berlin, 2009. URL: https://doi.org/10.1007/978-3-642-10631-6_107.
  17. Louis Ibarra. Finding pattern matchings for permutations. Inf. Process. Lett., 61(6):293-295, 1997. URL: https://doi.org/10.1016/S0020-0190(97)00029-X.
  18. Vít Jelínek and Jan Kynčl. Hardness of permutation pattern matching. In Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 378-396. SIAM, Philadelphia, PA, 2017. URL: https://doi.org/10.1137/1.9781611974782.24.
  19. Vít Jelínek, Michal Opler, and Pavel Valtr. Generalized coloring of permutations. In 26th European Symposium on Algorithms, volume 112 of LIPIcs. Leibniz Int. Proc. Inform., pages Art. No. 50, 14. Schloss Dagstuhl. Leibniz-Zent. Inform., Wadern, 2018. Google Scholar
  20. Erkki Mäkinen. On the longest upsequence problem for permutations. Int. J. Comput. Math., 77(1):45-53, 2001. URL: https://doi.org/10.1080/00207160108805049.
  21. Adam Marcus and Gábor Tardos. Excluded permutation matrices and the Stanley-Wilf conjecture. J. Combin. Theory Ser. A, 107(1):153-160, 2004. URL: https://doi.org/10.1016/j.jcta.2004.04.002.
  22. Maximillian M. Murphy and Vincent R. Vatter. Profile classes and partial well-order for permutations. Electron. J. Combin., 9(2):Research paper 17, 30, 2002/03. Permutation patterns (Otago, 2003). URL: http://www.combinatorics.org/Volume_9/Abstracts/v9i2r17.html.
  23. Both Emerite Neou. Permutation pattern matching. PhD thesis, Université Paris-Est ; Université de Vérone (Italie), 2017. URL: https://pastel.archives-ouvertes.fr/tel-01866721.
  24. Both Emerite Neou, Romeo Rizzi, and Stéphane Vialette. Permutation pattern matching in (213, 231)-avoiding permutations. Discret. Math. Theor. Comput. Sci., 18(2), 2016. URL: http://dmtcs.episciences.org/3199.
  25. Vincent Vatter. Small permutation classes. Proc. Lond. Math. Soc. (3), 103(5):879-921, 2011. URL: https://doi.org/10.1112/plms/pdr017.
  26. Vincent Vatter. Finding regular insertion encodings for permutation classes. J. Symbolic Comput., 47(3):259-265, 2012. URL: https://doi.org/10.1016/j.jsc.2011.11.002.
  27. Vincent Vatter. Growth rates of permutation classes: from countable to uncountable. Proc. Lond. Math. Soc. (3), 119(4):960-997, 2019. URL: https://doi.org/10.1112/plms.12250.
  28. V. Yugandhar and Sanjeev Saxena. Parallel algorithms for separable permutations. Discret. Appl. Math., 146(3):343-364, 2005. URL: https://doi.org/10.1016/j.dam.2004.10.004.
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