# The Complexity of Pattern Matching for 321-Avoiding and Skew-Merged Permutations

@article{Albert2016TheCO, title={The Complexity of Pattern Matching for 321-Avoiding and Skew-Merged Permutations}, author={Michael H. Albert and Marie-Louise Lackner and Martin Lackner and Vincent Vatter}, journal={Discret. Math. Theor. Comput. Sci.}, year={2016}, volume={18} }

The Permutation Pattern Matching problem, asking whether a pattern permutation $\pi$ is contained in a permutation $\tau$, is known to be NP-complete. In this paper we present two polynomial time algorithms for special cases. The first algorithm is applicable if both $\pi$ and $\tau$ are $321$-avoiding; the second is applicable if $\pi$ and $\tau$ are skew-merged. Both algorithms have a runtime of $O(kn)$, where $k$ is the length of $\pi$ and $n$ the length of $\tau$.

## 18 Citations

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By designing two-way reductions that imply that an $O(n^{4/3-\varepsilon})$ time algorithm for counting occurrences would imply an exciting breakthrough for counting (and hence also detecting) 4-cycles, a systematic study of the complexity of counting occurrences for different patterns of fixed small length k is investigated.

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A new hardness reduction is presented which allows us to show, in a uniform way, that Av(σ)-PPM is hard for every σ of size at least 6, for everyπ of size 5 except the symmetry class of 41352, as well as forevery σ symmetric to one of the three permutations 4321, 4312 and 4231.

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The Doubly Partially Ordered Pattern Matching (or DPOP Matching) problem is studied, a natural extension of the Permutation pattern matching problem, and restrictions on several parameters/properties of the input are considered, giving a(n almost) complete landscape for the algorithmic complexity of the problem.

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For a hereditary permutation class $\mathcal{C}$, we say that two permutations $\pi$ and $\sigma$ of $\mathcal{C}$ are Wilf-equivalent in $\mathcal{C}$, if $\mathcal{C}$ has the same number of…

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Partially ordered patterns (POPs) generalize the notion of classical patterns studied in the literature in the context of permutations, words, compositions and partitions. In this paper, we give a…

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We prove that every proper subclass of the 321-avoiding permutations that is defined either by only finitely many additional restrictions or is well-quasi-ordered has a rational generating function.…

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We prove that every proper subclass of the 321-avoiding permutations that is defined either by only finitely many additional restrictions or is well-quasi-ordered has a rational generating function.…

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While the theory of labeled well-quasi-order has received significant attention in the graph setting, it has not yet been considered in the context of permutation patterns. We initiate this study…

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