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

The Permutation Pattern Matching problem, asking whether a pattern permutation $π$ is contained in a permutation $τ$, 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 $π$ and $τ$ are $321$-avoiding; the second is applicable if $π$ and $τ$ are skew-merged. Both algorithms have a runtime of $O(kn)$, where $k$ is the length of $π$ and $n$ the length of $τ$.