Kernelization Bounds for Constrained Coloring

TL;DR

This paper investigates the kernel complexity of certain constraint satisfaction and coloring problems, establishing bounds and conditions under which polynomial kernels are possible, with implications for various graph and hypergraph coloring scenarios.

Contribution

It provides new lower bounds on kernel sizes for CSPs with specific relations and determines kernel bounds for rainbow-free coloring problems under NP-hardness assumptions.

Findings

Conditional lower bounds on kernel size based on the largest OR relation arity.

Exact kernel size bounds for rainbow-free coloring problems for all parameter ranges.

Applications to graph and hypergraph coloring problems with nearly tight kernel bounds.

Abstract

We study the kernel complexity of constraint satisfaction problems over a finite domain, parameterized by the number of variables, whose constraint language consists of two relations: the non-equality relation and an additional permutation-invariant relation $R$. We establish a conditional lower bound on the kernel size in terms of the largest arity of an OR relation definable from $R$. Building on this, we investigate the kernel complexity of uniformly rainbow free coloring problems. In these problems, for fixed positive integers $d$, $\ell$, and $q \geq d$, we are given a graph $G$ on $n$ vertices and a collection $\cal F$ of $\ell$-tuples of $d$-subsets of its vertex set, and the goal is to decide whether there exists a proper coloring of $G$ with $q$ colors such that no $\ell$-tuple in $\cal F$ is uniformly rainbow, that is, no tuple has all its sets colored with the same $d$ distinct colors. We determine, for all admissible values of $d$, $\ell$, and $q$, the infimum over all values $\eta$ for which the problem admits a kernel of size $O(n^\eta)$, under the assumption $\mathsf{NP} \nsubseteq \mathsf{coNP/poly}$. As applications, we obtain nearly tight bounds on the kernel complexity of various coloring problems under diverse settings and parameterizations. This includes graph coloring problems parameterized by the vertex-deletion distance to a disjoint union of cliques, resolving a question of Schalken (2020), as well as uniform hypergraph coloring problems parameterized by the number of vertices, extending results of Jansen and Pieterse (2019) and Beukers (2021).