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Kernelization for Orthogonality Dimension

Authors Ishay Haviv , Dror Rabinovich

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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.

References

  1. Ziv Bar-Yossef, Yitzhak Birk, T. S. Jayram, and Tomer Kol. Index coding with side information. IEEE Trans. Inform. Theory, 57(3):1479-1494, 2011. Preliminary version in FOCS'06. URL: https://doi.org/10.1109/TIT.2010.2103753.
  2. Alexander Barg and Gilles Zémor. High-rate storage codes on triangle-free graphs. IEEE Trans. Inform. Theory, 68(12):7787-7797, 2022. URL: https://doi.org/10.1109/TIT.2022.3191309.
  3. Yitzhak Birk and Tomer Kol. Coding on demand by an informed source (ISCOD) for efficient broadcast of different supplemental data to caching clients. IEEE Trans. Inform. Theory, 52(6):2825-2830, 2006. Preliminary version in INFOCOM'98. URL: https://doi.org/10.1109/TIT.2006.874540.
  4. Hans L. Bodlaender, Rodney G. Downey, Michael R. Fellows, and Danny Hermelin. On problems without polynomial kernels. J. Comput. Syst. Sci., 75(8):423-434, 2009. Preliminary version in ICALP'08. URL: https://doi.org/10.1016/J.JCSS.2009.04.001.
  5. Hans L. Bodlaender, Bart M. P. Jansen, and Stefan Kratsch. Kernelization lower bounds by cross-composition. SIAM J. Discret. Math., 28(1):277-305, 2014. Preliminary version in STACS'11. URL: https://doi.org/10.1137/120880240.
  6. Jop Briët, Harry Buhrman, Debbie Leung, Teresa Piovesan, and Florian Speelman. Round elimination in exact communication complexity. In Proc. of the 10th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC'15), pages 206-225, 2015. URL: https://doi.org/10.4230/LIPICS.TQC.2015.206.
  7. Jop Briët and Jeroen Zuiddam. On the orthogonal rank of Cayley graphs and impossibility of quantum round elimination. Quantum Information & Computation, 17(1&2):106-116, 2017. Google Scholar
  8. Leizhen Cai. Parameterized complexity of vertex colouring. Discret. Appl. Math., 127(3):415-429, 2003. URL: https://doi.org/10.1016/S0166-218X(02)00242-1.
  9. Peter J. Cameron, Ashley Montanaro, Michael W. Newman, Simone Severini, and Andreas J. Winter. On the quantum chromatic number of a graph. Electron. J. Comb., 14(1):R81, 2007. URL: https://doi.org/10.37236/999.
  10. John F. Canny. Some algebraic and geometric computations in PSPACE. In Proc. of the 20th Annual ACM Symposium on Theory of Computing (STOC'88), pages 460-467, 1988. URL: https://doi.org/10.1145/62212.62257.
  11. Dror Chawin and Ishay Haviv. On the subspace choosability in graphs. Electron. J. Comb., 29(2):P2.20, 2022. Preliminary version in EuroComb'21. URL: https://doi.org/10.37236/10765.
  12. Dror Chawin and Ishay Haviv. Improved NP-hardness of approximation for orthogonality dimension and minrank. SIAM J. Discret. Math., 37(4):2670-2688, 2023. Preliminary version in STACS'23. URL: https://doi.org/10.1137/23M155760X.
  13. Bruno Codenotti, Pavel Pudlák, and Giovanni Resta. Some structural properties of low-rank matrices related to computational complexity. Theor. Comput. Sci., 235(1):89-107, 2000. URL: https://doi.org/10.1016/S0304-3975(99)00185-1.
  14. Marek Cygan, Fedor V. Fomin, Lukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michal Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 2015. URL: https://doi.org/10.1007/978-3-319-21275-3.
  15. Ronald de Wolf. Quantum Computing and Communication Complexity. Ph.D. Thesis, Universiteit van Amsterdam, 2001. Google Scholar
  16. Fedor V. Fomin, Bart M. P. Jansen, and Michal Pilipczuk. Preprocessing subgraph and minor problems: When does a small vertex cover help? J. Comput. Syst. Sci., 80(2):468-495, 2014. Preliminary version in IPEC'12. URL: https://doi.org/10.1016/J.JCSS.2013.09.004.
  17. Fedor V. Fomin, Daniel Lokshtanov, Saket Saurabh, and Meirav Zehavi. Kernelization: Theory of Parameterized Preprocessing. Cambridge University Press, 2019. Google Scholar
  18. Alexander Golovnev and Ishay Haviv. The (generalized) orthogonality dimension of (generalized) Kneser graphs: Bounds and applications. Theory of Computing, 18(22):1-22, 2022. Preliminary version in CCC'21. URL: https://theoryofcomputing.org/articles/v018a022/.
  19. Alexander Golovnev, Oded Regev, and Omri Weinstein. The minrank of random graphs. IEEE Trans. Inform. Theory, 64(11):6990-6995, 2018. Preliminary version in RANDOM'17. URL: https://doi.org/10.1109/TIT.2018.2810384.
  20. Ishay Haviv. Approximating the orthogonality dimension of graphs and hypergraphs. Chic. J. Theor. Comput. Sci., 2022(2):1-26, 2022. Preliminary version in MFCS'19. Google Scholar
  21. Gerald Haynes, Catherine Park, Amanda Schaeffer, Jordan Webster, and Lon H. Mitchell. Orthogonal vector coloring. Electron. J. Comb., 17(1):R55, 2010. URL: https://doi.org/10.37236/327.
  22. Bart M. P. Jansen. The Power of Data Reduction: Kernels for Fundamental Graph Problems. Ph.D. Thesis, Utrecht University, 2013. Google Scholar
  23. Bart M. P. Jansen and Stefan Kratsch. Data reduction for graph coloring problems. Inf. Comput., 231:70-88, 2013. Preliminary version in FCT'11. URL: https://doi.org/10.1016/J.IC.2013.08.005.
  24. Bart M. P. Jansen and Astrid Pieterse. Optimal data reduction for graph coloring using low-degree polynomials. Algorithmica, 81(10):3865-3889, 2019. Preliminary version in IPEC'17. URL: https://doi.org/10.1007/S00453-019-00578-5.
  25. Bart M. P. Jansen and Astrid Pieterse. Optimal sparsification for some binary CSPs using low-degree polynomials. ACM Trans. Comput. Theory, 11(4):28:1-26, 2019. Preliminary version in MFCS'16. URL: https://doi.org/10.1145/3349618.
  26. László Lovász. On the Shannon capacity of a graph. IEEE Trans. Inform. Theory, 25(1):1-7, 1979. URL: https://doi.org/10.1109/TIT.1979.1055985.
  27. László Lovász. Graphs and Geometry, volume 65. Colloquium Publications, 2019. Google Scholar
  28. László Lovász, Michael Saks, and Alexander Schrijver. Orthogonal representations and connectivity of graphs. Linear Algebra Appl., 114-115:439-454, 1989. Special Issue Dedicated to Alan J. Hoffman. Google Scholar
  29. René Peeters. Orthogonal representations over finite fields and the chromatic number of graphs. Combinatorica, 16(3):417-431, 1996. URL: https://doi.org/10.1007/BF01261326.
  30. Søren Riis. Information flows, graphs and their guessing numbers. Electron. J. Comb., 14(1):R44, 2007. Preliminary version in NETCOD'06. URL: https://doi.org/10.37236/962.
  31. Alfred Tarski. A Decision Method for Elementary Algebra and Geometry. Number R-109 in RAND Research & Commentary Reports. RAND, Santa Monica, CA, 1951. Google Scholar
  32. Chee-Keng Yap. Some consequences of non-uniform conditions on uniform classes. Theor. Comput. Sci., 26(3):287-300, 1983. URL: https://doi.org/10.1016/0304-3975(83)90020-8.
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