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 $4231$. 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.
翻译:我们研究了决定问题的复杂程度, 称为“ 变换模式匹配 ” 或 PPM 。 PPM 的输入由一对美元( text) 和 美元( ppen) 构成, 目标是确定$( tau) 是否包含$\ pi美元作为子变色。 在一般输入中, PPM 已知是 NP- 完成的, 是 Bose、 Busss 和 Lubiw 的结果 。 在本文中, 我们关注限量的 PPM 实例, 假设文本可以避免固定( 小) 模式 $( sigma$ ) ; 这一限制被称为 Av( $ ( grama美元) 和 $( PPPM ) 。 先前显示, Av( $( gmam) 美元) - PPM 中的任何大小, 以美元表示一个数字( 美元), 以每平价( 美元) 以每平价( 美元) 的方式, 以每平价( 美元) 平价( 美元) 美元) 美元的方式, 以每平价( 平价( 美元) 美元) 平价( 美元) 美元) 美元) 以每平价( 美元) 美元) 美元(美元) 美元) 美元(美元) 以平价( 美元) 美元) 美元) 美元(美元) 美元(美元) 的方式, 。