Dynamic Complexity Meets Parameterised Algorithms

TL;DR

This paper explores extending dynamic complexity classes with parameterised algorithms, allowing additional resources based on a parameter, and investigates their applicability through case studies on well-known problems.

Contribution

It introduces parameterised extensions of dynamic complexity classes and compares them with classical classes, applying methods from parameterised algorithms to dynamic settings.

Findings

Parameterised extensions of DynFO are defined and analyzed.

Methods from parameterised algorithms are applicable to dynamic complexity.

Case studies demonstrate the effectiveness of these extensions on known problems.

Abstract

Dynamic Complexity studies the maintainability of queries with logical formulas in a setting where the underlying structure or database changes over time. Most often, these formulas are from first-order logic, giving rise to the dynamic complexity class DynFO. This paper investigates extensions of DynFO in the spirit of parameterised algorithms. In this setting structures come with a parameter $k$ and the extensions allow additional "space" of size $f(k)$ (in the form of an additional structure of this size) or additional time $f(k)$ (in the form of iterations of formulas) or both. The resulting classes are compared with their non-dynamic counterparts and other classes. The main part of the paper explores the applicability of methods for parameterised algorithms to this setting through case studies for various well-known parameterised problems.