String editing under pattern constraints

We introduce the novel Nearest Pattern Constrained String (NPCS) problem of finding a minimum set $Q$ of character mutation, insertion, and deletion edit operations sufficient to modify a string x to contain all contiguous substrings in a pattern set $P$ and no contiguous substrings in a forbidden pattern set $F$. Letting Σ be the alphabet of allowed characters, and letting η and ϒ be the longest string length and sum of all string lengths in $P\cup F$, respectively, we show that NPCS is fixed-parameter tractable in $\left|P\right|$ with time complexity $O(2^{\left|P\right|}\cdot \Upsilon \cdot |\Sigma |\cdot \left(\right|P|+\eta )\left(\right|x|+1))$. Additionally, we consider a generalization of the NPCS problem in which we allow for constraints based on the membership of substrings in regular languages. In particular, we introduce a problem we denote String Editing under Substring in Language Constraints (StrEdit-SILC), where provided a wildcard-free string $x\in {\Sigma }^{⁎}$, a finite set of regular languages $R=\{L_1,L_2,\dots \}$, and a regular language $L_F$, the objective is to find a minimum cost set of mutation, insertion, and deletion edit operations $Q$ that suffice to convert the input string x into a string $x^{\prime }\in {\Sigma }^{⁎}$, where no substring has membership in $L_F$, and $\forall L_i\in R$, there exists a substring in $L_i$. Here, letting Ψ and ϖ be the sum of all regular expression lengths and longest regular expression length for languages in $R\cup \left\{L_F\right\}$, respectively, and letting $C_{mid}\in N$ be the maximum cost of an edit operation, we show that StrEdit-SILC is fixed-parameter tractable with respect to Ψ, having time complexity $O(2^{\Psi }\cdot |x|\cdot (\varpi \cdot |\Sigma |+C_{mid}))$. However, we also show that StrEdit-SILC is MAX-SNP-hard and otherwise difficult to approximate under stringent constraints.