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A Fast Algorithm for Permutation Pattern Matching Based on Alternating Runs

The NP-complete Permutation Pattern Matching problem asks whether a $k$-permutation $P$ is contained in a $n$-permutation $T$ as a pattern. This is the case if there exists an order-preserving embedding of $P$ into $T$. In this paper, we present a fixed-parameter algorithm solving this problem with a worst-case runtime of $\mathcal{O}(1.79^{\mathsf{run}(T)}\cdot n\cdot k)$, where $\mathsf{run}(T)$ denotes the number of alternating runs of $T$. This algorithm is particularly well-suited for instances where $T$ has few runs, i.e., few ups and downs. Moreover, since $\mathsf{run}(T)<n$, this can be seen as a $\mathcal{O}(1.79^{n}\cdot n\cdot k)$ algorithm which is the first to beat the exponential $2^n$ runtime of brute-force search. Furthermore, we prove that under standard complexity theoretic assumptions such a fixed-parameter tractability result is not possible for $\mathsf{run}(P)$.