Kernelization and Sparseness: the case of Dominating Set

TL;DR

This paper establishes new kernelization results for the Dominating Set problem on sparse graph classes, using graph theory concepts, and highlights complexity boundaries between different graph classes.

Contribution

It introduces a simplified, theory-based approach to kernelization for Dominating Set on classes of sparse graphs, extending previous results.

Findings

Linear kernels for r-Dominating Set on bounded expansion graphs

Almost linear kernels for Dominating Set on nowhere dense classes

W[2]-hardness of r-Dominating Set on somewhere dense classes

Abstract

We prove that for every positive integer $r$ and for every graph class $\mathcal G$ of bounded expansion, the $r$-Dominating Set problem admits a linear kernel on graphs from $\mathcal G$. Moreover, when $\mathcal G$ is only assumed to be nowhere dense, then we give an almost linear kernel on $\mathcal G$ for the classic Dominating Set problem, i.e., for the case $r=1$. These results generalize a line of previous research on finding linear kernels for Dominating Set and $r$-Dominating Set. However, the approach taken in this work, which is based on the theory of sparse graphs, is radically different and conceptually much simpler than the previous approaches. We complement our findings by showing that for the closely related Connected Dominating Set problem, the existence of such kernelization algorithms is unlikely, even though the problem is known to admit a linear kernel on $H$-topological-minor-free graphs. Also, we prove that for any somewhere dense class $\mathcal G$, there is some $r$ for which $r$-Dominating Set is W[$2$]-hard on $\mathcal G$. Thus, our results fall short of proving a sharp dichotomy for the parameterized complexity of $r$-Dominating Set on subgraph-monotone graph classes: we conjecture that the border of tractability lies exactly between nowhere dense and somewhere dense graph classes.