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  "path": "/abs/2604.21531v1",
  "publishedAt": "2026-04-24T00:00:00.000Z",
  "site": "https://arxiv.org",
  "tags": [
    "Ishay Haviv"
  ],
  "textContent": "**Authors:** Ishay Haviv\n\nWe 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).",
  "title": "Kernelization Bounds for Constrained Coloring"
}