On the Parameterised Complexity of String Morphism Problems

Abstract

Given a source string u and a target string w, to decide whether w can be obtained by applying a string morphism on u (i. e., uniformly replacing the symbols in u by strings) constitutes an NP-complete problem. For example, the target string w := baaba can be obtained from the source string u := aba, by replacing a and b in u by the strings ba and a, respectively. In this paper, we contribute to the recently started investigation of the computational complexity of the string morphism problem by studying it in the framework of parameterised complexity. ACM Subject Classification IntroductionMany of the typical string problems are concerned with special kinds of string operations like, e. g., concatenating strings with each other, deleting symbols from or inserting symbols into a string or replacing symbols by other symbols or even by other strings. Among the most prominent of these string problems are string-to-string correction, sequence alignment as well as the longest common subsequence and shortest common supersequence problem. The complexity of these problems have been intensely studied, both in the classical sense as well as in the parameterised setting (see, e. g., [1,9]). In this work, we investigate string problems that arise from a less well-known operation on strings, i. e., mapping a source string u to a target string w by uniformly (i. e., by a mapping) replacing the symbols of u by strings. For example, we can turn the source string u := abba into the target string w := bbaaaaabba by replacing a and b of u by the strings bba and aa, respectively. On the other hand, w := abaaaaaabb cannot be obtained from u in a similar way. The string morphism problem (denoted by StrMorph) is to decide for two given strings u and w, whether or not w can be obtained from u by this kind of operation. Due to its simple definition, variants of this N P-complete problem can be found in many different areas of theoretical computer science. In fact, many respective results are scattered throughout the literature without pointers to each other and consulting the existing literature suggests that variants of the string morphism problem have emerged and have been investigated in different contexts without knowledge of other related work.

Document Spanners: From Expressive Power to Decision Problems

“…It was first observed by Stephan et al [34] that the membership problem for pattern languages is W[1]-complete if the number of variable occurrences (not of variables) is used as a parameter (see Fernau et al [11] for the full proof). As the number of variable occurrences in a pattern corresponds to the number of variables in an equivalent spanner, this implies that using the number of variables in a spanner as parameter leads to W[1]-hardness for this parameter of CSp−Eval.…”

“…Fernau and Schmid [10] and Fernau et al [11] discuss these and various other potential restrictions to pattern languages that still do not lead to tractability (among these a bound on the length of the replacement of each variable, which corresponds to a bound on the length of spans). On the other hand, Reidenbach and Schmid [33] and Fernau et al [9] examine parameters for patterns that make the membership problem tractable.…”

Closure Properties of Pattern Languages

“…Hence, the knowledge on pattern languages is still patchy, despite recent progress mainly regarding decision problems (see, e. g., Freydenberger, Reidenbach [5], Fernau, Schmid [3], Fernau et al [4] and Reidenbach, Schmid [13]) and the relation to the Chomsky hierarchy (see Jain et al [6] and Reidenbach, Schmid [14]). …”

A Parameterized Study of Maximum Generalized Pattern Matching Problems

“…Given a pattern p and a text t, the goal is to find a mapping from the letters of p to non-empty substrings of t, such that applying the mapping to p results in t. Very recently, the problem has been investigated within the framework of parameterized complexity [Fernau, Schmid, and Villanger, 2013].…”

“…Given a pattern p and a text t, the goal is to find a mapping from the letters of p to non-empty substrings of t, such that applying the mapping to p results in t. Very recently, the problem has been investigated within the framework of parameterized complexity [Fernau, Schmid, and Villanger, 2013].In this paper we study the parameterized complexity of the optimization variant of GFM (called Max-GFM), which has been introduced in [Amir and Nor, 2007]. Here, one is allowed to replace some of the pattern letters with some special symbols "?…”