On the Parameterised Complexity of String Morphism Problems

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 -complete problem. We present a multivariate analysis of this problem (and its many variants) from the viewpoint of parameterised complexity theory, thereby pinning down the sources of its computational hardness. Our results show that most parameterised variants of the string morphism problem are fixed-parameter intractable and, apart from some very special cases, tractable variants can only be obtained by considering a large part of the input as parameters, namely the length of w and the number of different symbols in u.