Discontinuities in pattern inference

In the context of this paper, a pattern is a string that con- tains variables and terminals. A pattern α matches a terminal word w if w can be obtained by uniformly substituting the variables of α by termi- nal words. It is a well-known fact that deciding whether a given terminal word matches a given pattern is an NP-complete problem. In this work, we consider numerous parameters of this problem and for all possible combinations of these parameters, we investigate the question whether or not the variant obtained by bounding these parameters by constants can be solved ef fi ciently.

Pattern Matching with Variables: A Multivariate Complexity Analysis

Morphically primitive words

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 ����$\mathcal {NP}$-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.

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On the Parameterised Complexity of String Morphism Problems

A pattern α, i.e., a string that contains variables and terminals, matches a terminal word w if w can be obtained by uniformly substituting the variables of α by terminal words. Deciding whether a given terminal word matches a given pattern is NP-complete and this holds for several natural variants of the problem that result from whether or not variables can be erased, whether or not the patterns are required to be terminal-free or whether or not the mapping of variables to terminal words must be injective. We consider numerous parameters of this problem (i.e., number of variables, length of w, length of the words substituted for variables, number of occurrences per variable, cardinality of the terminal alphabet) and for all possible combinations of the parameters (and variants described above), we answer the question whether or not the problem is still NP-complete if these parameters are bounded by constants.

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Andrzej Ehrenfreucht

Papers

Finding a homomorphism between two words in NP-complete