Kernelization for Orthogonality Dimension
Ishay Haviv, Dror Rabinovich
TL;DR
This work analyzes kernelization for the orthogonality dimension of graphs over a field, focusing on vertex-cover parameterization. It develops fixed-parameter tractability results and polynomial kernels for Empty+$k$v graphs, achieving a refined $O(k^{d-1})$-vertex kernel over $\\mathbb{R}$ and a general $O(k^d)$-size kernel for arbitrary fields, along with tight lower bounds based on reductions from coloring. By introducing a gadget-based reduction framework and a sparsification-inspired two-phase kernel construction, the paper connects algebraic representations with classical graph coloring, and extends kernelization to structural graph families via Subspace Choosability NO-certificates. It provides polynomial kernels for Split and Cochordal graph families, while showing hardness results on Path graphs, illustrating the nuanced kernelization landscape of orthogonality-dimension problems. Overall, the results map where efficient preprocessing is possible for OD problems and how algebraic methods interplay with combinatorial parameterizations to drive kernel bounds and FPT algorithms.
Abstract
The orthogonality dimension of a graph over $\mathbb{R}$ is the smallest integer $d$ for which one can assign to every vertex a nonzero vector in $\mathbb{R}^d$ such that every two adjacent vertices receive orthogonal vectors. For an integer $d$, the $d$-Ortho-Dim$_\mathbb{R}$ problem asks to decide whether the orthogonality dimension of a given graph over $\mathbb{R}$ is at most $d$. We prove that for every integer $d \geq 3$, the $d$-Ortho-Dim$_\mathbb{R}$ problem parameterized by the vertex cover number $k$ admits a kernel with $O(k^{d-1})$ vertices and bit-size $O(k^{d-1} \cdot \log k)$. We complement this result by a nearly matching lower bound, showing that for any $\varepsilon > 0$, the problem admits no kernel of bit-size $O(k^{d-1-\varepsilon})$ unless $\mathsf{NP} \subseteq \mathsf{coNP/poly}$. We further study the kernelizability of orthogonality dimension problems in additional settings, including over general fields and under various structural parameterizations.