🤖 AI Summary
This work investigates the kernelization complexity of constraint satisfaction problems over finite domains with disequalities and permutation-invariant relations, parameterized by the number of variables, as well as the existence of polynomial kernels for the Uniform Rainbow Independent Set problem. By characterizing the largest arity OR relation definable via a given relation R and leveraging tools from parameterized complexity, kernelization techniques, logical definability, and the conditional assumption that NP ⊈ coNP/poly, the study establishes the first optimal lower bounds on kernel size for Uniform Rainbow Independent Set across all relevant parameterizations. It also resolves an open question regarding vertex-deletion distance to disjoint unions of cliques in graph coloring. The results yield nearly tight upper and lower bounds on the kernel complexity for several graph and hypergraph coloring problems, unifying and extending prior work by Jansen–Pieterse and Beukers et al.
📝 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 $η$ for which the problem admits a kernel of size $O(n^η)$, 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).