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Generalized Coloring of Permutations

Authors Vít Jelínek , Michal Opler , Pavel Valtr

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Author Details

Vít Jelínek
  • Computer Science Institute, Charles University, Malostranské náměstí 25, Praha 1, 11800, Czechia
Michal Opler
  • Computer Science Institute, Charles University, Malostranské náměstí 25, Praha 1, 11800, Czechia
Pavel Valtr
  • Department of Applied Mathematics, Charles University, Malostranské náměstí 25, Praha 1, 11800, Czechia

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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)


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.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Permutations and combinations
  • Mathematics of computing → Combinatorial algorithms
  • Theory of computation → Problems, reductions and completeness
  • Permutations
  • merge
  • generalized coloring


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