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

TL;DR

This paper presents polynomial time algorithms for the permutation pattern matching problem in special cases where permutations are either 321-avoiding or skew-merged, improving understanding of these specific instances.

Contribution

It introduces two efficient algorithms with O(kn) runtime for pattern matching in 321-avoiding and skew-merged permutations, addressing NP-complete cases.

Findings

Algorithms run in O(kn) time for both cases

Efficient solutions for special permutation classes

Advances understanding of permutation pattern matching complexity

Abstract

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 $\tau$.