A Parameterized Study of Maximum Generalized Pattern Matching Problems

TL;DR

This paper conducts a comprehensive parameterized complexity analysis of the Max-GFM problem, an optimization variant of generalized function matching involving wildcards, providing a complete classification under various parameters.

Contribution

It offers the first complete classification of the parameterized complexity of Max-GFM and its variants across multiple parameters, advancing understanding of this problem's computational boundaries.

Findings

Complexity classifications for various parameterizations.

Identification of fixed-parameter tractable cases.

Demonstration of hardness results for other parameters.

Abstract

The generalized function matching (GFM) problem has been intensively studied starting with [Ehrenfeucht and Rozenberg, 1979]. 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 "?", termed wildcards or don't cares, which can be mapped to an arbitrary substring of the text. The goal is to minimize the number of wildcards used. We give a complete classification of the parameterized complexity of Max-GFM and its variants under a wide range of parameterizations, such as, the number of occurrences of a letter in the text, the size of the text alphabet, the number of occurrences of a letter in the pattern, the size of the pattern alphabet, the maximum length of a string matched to any pattern letter, the number of wildcards and the maximum size of a string that a wildcard can be mapped to.