Completeness Results for Parameterized Space Classes

TL;DR

This paper establishes the completeness of specific problems for parameterized space classes, advancing the understanding of their complexity within the parameterized hierarchy.

Contribution

It proves the associative generability and longest common subsequence problems are complete for parameterized space classes, introducing a union operation as a technical tool.

Findings

Associative generability problem is complete for parameterized space classes.

Longest common subsequence problem is complete for parameterized space classes.

Introduces a union operation linking classical and W-class complexity results.

Abstract

The parameterized complexity of a problem is considered "settled" once it has been shown to lie in FPT or to be complete for a class in the W-hierarchy or a similar parameterized hierarchy. Several natural parameterized problems have, however, resisted such a classification. At least in some cases, the reason is that upper and lower bounds for their parameterized space complexity have recently been obtained that rule out completeness results for parameterized time classes. In this paper, we make progress in this direction by proving that the associative generability problem and the longest common subsequence problem are complete for parameterized space classes. These classes are defined in terms of different forms of bounded nondeterminism and in terms of simultaneous time--space bounds. As a technical tool we introduce a "union operation" that translates between problems complete for classical complexity classes and for W-classes.