Finding and counting permutations via CSPs

Benjamin Aram Berendsohn; László Kozma; Dániel Marx

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arXiv · 1908.04673

Finding and counting permutations via CSPs

Benjamin Aram Berendsohn·László Kozma·Dániel Marx

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Abstract

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 $f(k) \cdot n^{o(k/\log{k})}$ 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.

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Benjamin Aram Berendsohn, László Kozma, Dániel Marx. 2019-08-13. Finding and counting permutations via CSPs. https://arxiv.org/abs/1908.04673

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