We study the complexity of the decision problem known as Permutation Pattern Matching, or PPM. The input of PPM consists of a pair of permutations $\tau$ (the `text') and $\pi$ (the `pattern'), and the goal is to decide whether $\tau$ contains $\pi$ as a subpermutation. On general inputs, PPM is known to be NP-complete by a result of Bose, Buss and Lubiw. In this paper, we focus on restricted instances of PPM where the text is assumed to avoid a fixed (small) pattern $\sigma$; this restriction is known as Av($\sigma$)-PPM. It has been previously shown that Av($\sigma$)-PPM is polynomial for any $\sigma$ of size at most 3, while it is NP-hard for any $\sigma$ containing a monotone subsequence of length four. In this paper, we present a new hardness reduction which allows us to show, in a uniform way, that Av($\sigma$)-PPM is hard for every $\sigma$ of size at least 6, for every $\sigma$ of size 5 except the symmetry class of $41352$, as well as for every $\sigma$ symmetric to one of the three permutations $4321$, $4312$ and $4213$. Moreover, assuming the exponential time hypothesis, none of these hard cases of Av($\sigma$)-PPM can be solved in time $2^{o(n/\log n)}$. Previously, such conditional lower bound was not known even for the unconstrained PPM problem. On the tractability side, we combine the CSP approach of Guillemot and Marx with the structural results of Huczynska and Vatter to show that for any monotone-griddable permutation class C, PPM is polynomial when the text is restricted to a permutation from C.

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