Tight Kernel Bounds for Problems on Graphs with Small Degeneracy

Abstract

Kernelization is a strong and widely-applied technique in parameterized complexity. In a nutshell, a kernelization algorithm for a parameterized problem transforms a given instance of the problem into an equivalent instance whose size depends solely on the parameter. Recent years have seen major advances in the study of both upper and lower bound techniques for kernelization, and by now this area has become one of the major research threads in parameterized complexity.

We consider kernelization for problems on d-degenerate graphs, i.e. graphs such that any subgraph contains a vertex of degree at most d. This graph class generalizes many classes of graphs for which effective kernelization is known to exist, e.g. planar graphs, H-minor free graphs, H-topological minor free graphs. We show that for several natural problems on d-degenerate graphs the best known kernelization upper bounds are essentially tight. In particular, using intricate constructions of weak compositions, we prove that unless NP ⊆ coNP/poly:

• Dominating Set has no kernels of size O(k ^{(d − 1)(d − 3) − ε}) for any ε > 0. The current best upper bound is $O(k^{(d+1)^2})$.

• Independent Dominating Set has no kernels of size O(k ^{d − 4 − ε}) for any ε > 0. The current best upper bound is O(k ^d + 1).

• Induced Matching has no kernels of size O(k ^{d − 3 − ε}) for any ε > 0. The current best upper bound is O(k ^d).

We also give simple kernels for Connected Vertex Cover and Capacitated Vertex Cover of size O(k ^d) and O(k ^d + 1) respectively. Both these problems do not have kernels of size O(k ^{d − 1 − ε}) unless coNP/poly.

In this extended abstract we will focus on the lower bound for Dominating Set, which we feel is the central result of our study. The proofs of the other results can be found in the full version of the paper.

Partially supported by ERC Starting Grant NEWNET 279352.

A full version of the paper can be found in [8].