Levelable Sets and the Algebraic Structure of Parameterizations

TL;DR

This paper explores the algebraic structure of parameterizations in fixed-parameter tractability, revealing a hierarchy of complexity and showing that no practically fixed-parameter tractable sets have optimal parameterizations.

Contribution

It introduces an algebraic framework for parameterizations, enabling comparison and hierarchy establishment beyond traditional parameterized algorithm analysis.

Findings

Established a hierarchy of parameterizations based on algebraic structure

Proved no practically fixed-parameter tractable sets possess optimal parameterizations

Provided a new perspective on complexity measures in parameterized complexity

Abstract

Asking which sets are fixed-parameter tractable for a given parameterization constitutes much of the current research in parameterized complexity theory. This approach faces some of the core difficulties in complexity theory. By focussing instead on the parameterizations that make a given set fixed-parameter tractable, we circumvent these difficulties. We isolate parameterizations as independent measures of complexity and study their underlying algebraic structure. Thus we are able to compare parameterizations, which establishes a hierarchy of complexity that is much stronger than that present in typical parameterized algorithms races. Among other results, we find that no practically fixed-parameter tractable sets have optimal parameterizations.