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Kernelization for Orthogonality Dimension

Authors Ishay Haviv , Dror Rabinovich

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LIPIcs.IPEC.2024.8.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
  • Mathematics of computing → Graph coloring
  • Mathematics of computing → Combinatorial algorithms
Keywords
  • Orthogonality dimension
  • Fixed-parameter tractability
  • Kernelization
  • Graph coloring

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Abstract

The orthogonality dimension of a graph over ℝ is the smallest integer d for which one can assign to every vertex a nonzero vector in ℝ^d such that every two adjacent vertices receive orthogonal vectors. For an integer d, the d-Ortho-Dim_ℝ problem asks to decide whether the orthogonality dimension of a given graph over ℝ is at most d. We prove that for every integer d ≥ 3, the d-Ortho-Dim_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} ⋅ log k). We complement this result by a nearly matching lower bound, showing that for any ε > 0, the problem admits no kernel of bit-size O(k^{d-1-ε}) unless NP ⊆ coNP/poly. We further study the kernelizability of orthogonality dimension problems in additional settings, including over general fields and under various structural parameterizations.

Cite As Get BibTex

Ishay Haviv and Dror Rabinovich. Kernelization for Orthogonality Dimension. In 19th International Symposium on Parameterized and Exact Computation (IPEC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 321, pp. 8:1-8:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.IPEC.2024.8

Author Details

Ishay Haviv
  • School of Computer Science, The Academic College of Tel Aviv-Yaffo, Israel
Dror Rabinovich
  • School of Computer Science, The Academic College of Tel Aviv-Yaffo, Israel

Funding

  • Haviv, Ishay: Research supported by the Israel Science Foundation (grant No. 1218/20).
  • Rabinovich, Dror: Research supported by the Israel Science Foundation (grant No. 1218/20).

Acknowledgements

We are grateful to the anonymous reviewers for their constructive and helpful feedback.

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