Kernelization for Orthogonality Dimension

TL;DR

This paper investigates the kernelization complexity of the orthogonality dimension problem in graphs, providing polynomial kernels for fixed dimensions and establishing tight lower bounds, with implications for related settings.

Contribution

It introduces polynomial kernels for the orthogonality dimension problem parameterized by vertex cover size for all fixed dimensions $d \\geq 3$, and proves nearly matching lower bounds.

Findings

Polynomial kernels of size $O(k^{d-1})$ vertices for fixed $d \\geq 3$

Lower bounds showing no smaller kernels unless complexity collapses

Extension of kernelization results to other fields and parameters

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.