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# Permutation Pattern Matching for Doubly Partially Ordered Patterns

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## Cite As

Laurent Bulteau, Guillaume Fertin, Vincent Jugé, and Stéphane Vialette. Permutation Pattern Matching for Doubly Partially Ordered Patterns. In 33rd Annual Symposium on Combinatorial Pattern Matching (CPM 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 223, pp. 21:1-21:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.CPM.2022.21

## Abstract

We study in this paper the Doubly Partially Ordered Pattern Matching (or DPOP Matching) problem, a natural extension of the Permutation Pattern Matching problem. Permutation Pattern Matching takes as input two permutations σ and π, and asks whether there exists an occurrence of σ in π; whereas DPOP Matching takes two partial orders P_v and P_p defined on the same set X and a permutation π, and asks whether there exist |X| elements in π whose values (resp., positions) are in accordance with P_v (resp., P_p). Posets P_v and P_p aim at relaxing the conditions formerly imposed by the permutation σ, since σ yields a total order on both positions and values. Our problem being NP-hard in general (as Permutation Pattern Matching is), we consider restrictions on several parameters/properties of the input, e.g., bounding the size of the pattern, assuming symmetry of the posets (i.e., P_v and P_p are identical), assuming that one partial order is a total (resp., weak) order, bounding the length of the longest chain/anti-chain in the posets, or forbidding specific patterns in π. For each such restriction, we provide results which together give a(n almost) complete landscape for the algorithmic complexity of the problem.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Design and analysis of algorithms
##### Keywords
• Partial orders
• Permutations
• Pattern Matching
• Algorithmic Complexity
• Parameterized Complexity

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## References

1. Shlomo Ahal and Yuri Rabinovich. On complexity of the subpattern problem. SIAM J. Discret. Math., 22(2):629-649, 2008.
2. Michael H. Albert, Marie-Louise Lackner, Martin Lackner, and Vincent Vatter. The complexity of pattern matching for 321-avoiding and skew-merged permutations. Discret. Math. Theor. Comput. Sci., 18(2), 2016.
3. Sergey V. Avgustinovich, Sergey Kitaev, and Alexandr Valyuzhenich. Avoidance of boxed mesh patterns on permutations. Discret. Appl. Math., 161(1-2):43-51, 2013.
4. Eric Babson and Einar Steingrímsson. Generalized permutation patterns and a classification of the mahonian statistics. Séminaire Lotharingien de Combinatoire [electronic only], 44:B44b, 18 p.-B44b, 18 p., 2000. URL: http://eudml.org/doc/120841.
5. Benjamin Aram Berendsohn, László Kozma, and Dániel Marx. Finding and counting permutations via csps. Algorithmica, 83(8):2552-2577, 2021.
6. Miklós Bóna. Combinatorics of Permutations, Second Edition. Discrete mathematics and its applications. CRC Press, 2012.
7. Prosenjit Bose, Jonathan F. Buss, and Anna Lubiw. Pattern matching for permutations. Inf. Process. Lett., 65(5):277-283, 1998.
8. Mireille Bousquet-Mélou, Anders Claesson, Mark Dukes, and Sergey Kitaev. (2+2)-free posets, ascent sequences and pattern avoiding permutations. J. Comb. Theory, Ser. A, 117(7):884-909, 2010.
9. Petter Brändén and Anders Claesson. Mesh patterns and the expansion of permutation statistics as sums of permutation patterns. Electron. J. Comb., 18(2), 2011.
10. Marie-Louise Bruner and Martin Lackner. A fast algorithm for permutation pattern matching based on alternating runs. Algorithmica, 75(1):84-117, 2016.
11. Laurent Bulteau, Romeo Rizzi, and Stéphane Vialette. Pattern matching for k-track permutations. In Costas S. Iliopoulos, Hon Wai Leong, and Wing-Kin Sung, editors, Combinatorial Algorithms - 29th International Workshop, IWOCA 2018, Singapore, July 16-19, 2018, Proceedings, volume 10979 of Lecture Notes in Computer Science, pages 102-114. Springer, 2018.
12. Maxime Crochemore and Ely Porat. Fast computation of a longest increasing subsequence and application. Inf. Comput., 208(9):1054-1059, 2010.
13. Sergi Elizalde and Marc Noy. Consecutive patterns in permutations. Adv. Appl. Math., 30(1-2):110-125, 2003.
14. Paul Erdös and George Szekeres. A combinatorial problem in geometry. Compositio mathematica, 2:463-470, 1935.
15. Jacob Fox. Stanley-wilf limits are typically exponential. CoRR, abs/1310.8378, 2013. URL: http://arxiv.org/abs/1310.8378.
16. Paweł Gawrychowski and Mateusz Rzepecki. Faster exponential algorithm for permutation pattern matching. In Symposium on Simplicity in Algorithms (SOSA), pages 279-284. SIAM, 2022.
17. Sylvain Guillemot and Dániel Marx. Finding small patterns in permutations in linear time. In Chandra Chekuri, editor, Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014, Portland, Oregon, USA, January 5-7, 2014, pages 82-101. SIAM, 2014.
18. Sylvain Guillemot and Stéphane Vialette. Pattern matching for 321-avoiding permutations. In Yingfei Dong, Ding-Zhu Du, and Oscar H. Ibarra, editors, Algorithms and Computation, 20th International Symposium, ISAAC 2009, Honolulu, Hawaii, USA, December 16-18, 2009. Proceedings, volume 5878 of Lecture Notes in Computer Science, pages 1064-1073. Springer, 2009.
19. Louis Ibarra. Finding pattern matchings for permutations. Inf. Process. Lett., 61(6):293-295, 1997.
20. Klaus Jansen, Stefan Kratsch, Dániel Marx, and Ildikó Schlotter. Bin packing with fixed number of bins revisited. J. Comput. Syst. Sci., 79(1):39-49, 2013.
21. Vít Jelínek and Jan Kynčl. Hardness of permutation pattern matching. In Philip N. Klein, editor, Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017, Barcelona, Spain, Hotel Porta Fira, January 16-19, pages 378-396. SIAM, 2017.
22. Richard M. Karp. Reducibility among Combinatorial Problems, pages 85-103. Springer US, Boston, MA, 1972.
23. Sergey Kitaev. Introduction to partially ordered patterns. Discret. Appl. Math., 155(8):929-944, 2007.
24. Sergey Kitaev. Patterns in Permutations and Words. Monographs in Theoretical Computer Science. An EATCS Series. Springer, 2011.
25. Both Emerite Neou, Romeo Rizzi, and Stéphane Vialette. Pattern matching for separable permutations. In Shunsuke Inenaga, Kunihiko Sadakane, and Tetsuya Sakai, editors, String Processing and Information Retrieval - 23rd International Symposium, SPIRE 2016, Beppu, Japan, October 18-20, 2016, Proceedings, volume 9954 of Lecture Notes in Computer Science, pages 260-272, 2016.
26. Both Emerite Neou, Romeo Rizzi, and Stéphane Vialette. Permutation pattern matching in (213, 231)-avoiding permutations. Discret. Math. Theor. Comput. Sci., 18(2), 2016.