A W[1]-Completeness Result for Generalized Permutation Pattern Matching

TL;DR

This paper proves that the generalized permutation pattern matching problem is W[1]-complete when parameterized by pattern length, indicating it is unlikely to have fixed-parameter tractable algorithms based solely on this parameter.

Contribution

It establishes the W[1]-completeness of the generalized permutation pattern matching problem, extending known complexity results to a more general setting.

Findings

W[1]-completeness with respect to pattern length

No fixed-parameter tractable algorithms likely exist for this problem

Extends complexity understanding of permutation pattern matching

Abstract

The NP-complete Permutation Pattern Matching problem asks whether a permutation P (the pattern) can be matched into a permutation T (the text). A matching is an order-preserving embedding of P into T. In the Generalized Permutation Pattern Matching problem one can additionally enforce that certain adjacent elements in the pattern must be mapped to adjacent elements in the text. This paper studies the parameterized complexity of this more general problem. We show W[1]-completeness with respect to the length of the pattern P. Under standard complexity theoretic assumptions this implies that no fixed-parameter tractable algorithm can be found for any parameter depending solely on P.