Pattern matching with variables: Efficient algorithms and complexity results

A pattern α (i. e., a string of variables and terminals) matches a word w, if w can be obtained by uniformly replacing the variables of α by terminal words. The respective matching problem, i. e., deciding whether or not a given pattern matches a given word, is generally NP-complete, but can be solved in polynomial-time for restricted classes of patterns. We present efficient algorithms for the matching problem with respect to patterns with a bounded number of repeated variables and patterns with a structural restriction on the order of variables. Furthermore, we show that it is NP-complete to decide, for a given number k and a word w, whether w can be factorised into k distinct factors. As an immediate consequence of this hardness result, the injective version (i. e., different variables are replaced by different words) of the matching problem is NP-complete even for very restricted clases of patterns

Revisiting Shinohara's algorithm for computing descriptive patterns

A pattern α is a word consisting of constants and variables and it describes the pattern language L(α) of all words that can be obtained by uniformly replacing the variables with constant words. In 1982, Shinohara presents an algorithm that computes a pattern that is descriptive for a finite set S of words, i.e., its pattern language contains S in the closest possible way among all pattern languages. We generalise Shinohara’s algorithm to subclasses of patterns and characterise those subclasses for which it is applicable. Furthermore, within this set of pattern classes, we characterise those for which Shinohara’s algorithm has a polynomial running time (under the assumption P 6= N P). Moreover, we also investigate the complexity of the consistency problem of patterns, i.e., finding a pattern that separates two given finite sets of words