Polynomial Kernels and Wideness Properties of Nowhere Dense Graph Classes

TL;DR

This paper establishes polynomial bounds linking nowhere dense graph classes and uniform quasi-wideness using logic tools, enabling polynomial kernels for the distance-r dominating set problem.

Contribution

It introduces polynomial bounds for the equivalence of nowhere denseness and uniform quasi-wideness, and derives polynomial kernels for the distance-r dominating set problem on these classes.

Findings

Polynomial bounds for equivalence of nowhere denseness and uniform quasi-wideness.

Polynomial kernels for the distance-r dominating set problem on nowhere dense classes.

Implication that kernel existence for all r is equivalent to polynomial kernels for all r under standard complexity assumptions.

Abstract

Nowhere dense classes of graphs are very general classes of uniformly sparse graphs with several seemingly unrelated characterisations. From an algorithmic perspective, a characterisation of these classes in terms of uniform quasi-wideness, a concept originating in finite model theory, has proved to be particularly useful. Uniform quasi-wideness is used in many fpt-algorithms on nowhere dense classes. However, the existing constructions showing the equivalence of nowhere denseness and uniform quasi-wideness imply a non-elementary blow up in the parameter dependence of the fpt-algorithms, making them infeasible in practice. As a first main result of this paper, we use tools from logic, in particular from a subfield of model theory known as stability theory, to establish polynomial bounds for the equivalence of nowhere denseness and uniform quasi-wideness. A powerful method in parameterized complexity theory is to compute a problem kernel in a pre-computation step, that is, to reduce the input instance in polynomial time to a sub-instance of size bounded in the parameter only (independently of the input graph size). Our new tools allow us to obtain for every fixed value of $r$ a polynomial kernel for the distance-$r$ dominating set problem on nowhere dense classes of graphs. This result is particularly interesting, as it implies that for every class $\mathcal{C}$ of graphs which is closed under subgraphs, the distance-$r$ dominating set problem admits a kernel on $\mathcal{C}$ for every value of $r$ if, and only if, it admits a polynomial kernel for every value of $r$ (under the standard assumption of parameterized complexity theory that $\mathrm{FPT} \neq W[2]$).