License: Creative Commons Attribution 3.0 Unported license (CC-BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.IPEC.2019.1
URN: urn:nbn:de:0030-drops-114627
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Berendsohn, Benjamin Aram ; Kozma, László ; Marx, Dániel

Finding and Counting Permutations via CSPs

LIPIcs-IPEC-2019-1.pdf (0.5 MB)


Permutation patterns and pattern avoidance have been intensively studied in combinatorics and computer science, going back at least to the seminal work of Knuth on stack-sorting (1968). Perhaps the most natural algorithmic question in this area is deciding whether a given permutation of length n contains a given pattern of length k. In this work we give two new algorithms for this well-studied problem, one whose running time is n^{k/4 + o(k)}, and a polynomial-space algorithm whose running time is the better of O(1.6181^n) and O(n^{k/2 + 1}). These results improve the earlier best bounds of n^{0.47k + o(k)} and O(1.79^n) due to Ahal and Rabinovich (2000) resp. Bruner and Lackner (2012) and are the fastest algorithms for the problem when k in Omega(log{n}). We show that both our new algorithms and the previous exponential-time algorithms in the literature can be viewed through the unifying lens of constraint-satisfaction. Our algorithms can also count, within the same running time, the number of occurrences of a pattern. We show that this result is close to optimal: solving the counting problem in time f(k) * n^{o(k/log{k})} would contradict the exponential-time hypothesis (ETH). For some special classes of patterns we obtain improved running times. We further prove that 3-increasing and 3-decreasing permutations can, in some sense, embed arbitrary permutations of almost linear length, which indicates that an algorithm with sub-exponential running time is unlikely, even for patterns from these restricted classes.

BibTeX - Entry

  author =	{Benjamin Aram Berendsohn and L{\'a}szl{\'o} Kozma and D{\'a}niel Marx},
  title =	{{Finding and Counting Permutations via CSPs}},
  booktitle =	{14th International Symposium on Parameterized and Exact Computation (IPEC 2019)},
  pages =	{1:1--1:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-129-0},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{148},
  editor =	{Bart M. P. Jansen and Jan Arne Telle},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{},
  URN =		{urn:nbn:de:0030-drops-114627},
  doi =		{10.4230/LIPIcs.IPEC.2019.1},
  annote =	{Keywords: permutations, pattern matching, constraint satisfaction, exponential time}

Keywords: permutations, pattern matching, constraint satisfaction, exponential time
Collection: 14th International Symposium on Parameterized and Exact Computation (IPEC 2019)
Issue Date: 2019
Date of publication: 04.12.2019

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