Optimal Data Reduction for Graph Coloring Using Low-Degree Polynomials

Authors Bart M. P. Jansen, Astrid Pieterse

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Bart M. P. Jansen
Astrid Pieterse

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Bart M. P. Jansen and Astrid Pieterse. Optimal Data Reduction for Graph Coloring Using Low-Degree Polynomials. In 12th International Symposium on Parameterized and Exact Computation (IPEC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 89, pp. 22:1-22:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)


The theory of kernelization can be used to rigorously analyze data reduction for graph coloring problems. Here, the aim is to reduce a q-Coloring input to an equivalent but smaller input whose size is provably bounded in terms of structural properties, such as the size of a minimum vertex cover. In this paper we settle two open problems about data reduction for q-Coloring. First, we use a recent technique of finding redundant constraints by representing them as low-degree polynomials, to obtain a kernel of bitsize O(k^(q-1) log k) for q-Coloring parameterized by Vertex Cover for any q >= 3. This size bound is optimal up to k^o(1) factors assuming NP is not a subset of coNP/poly, and improves on the previous-best kernel of size O(k^q). Our second result shows that 3-Coloring does not admit non-trivial sparsification: assuming NP is not a subset of coNP/poly, the parameterization by the number of vertices n admits no (generalized) kernel of size O(n^(2-e)) for any e > 0. Previously, such a lower bound was only known for coloring with q >= 4 colors.
  • graph coloring
  • kernelization
  • sparsification


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  1. Hans L. Bodlaender, Bart M. P. Jansen, and Stefan Kratsch. Kernelization lower bounds by cross-composition. SIAM J. Discrete Math., 28(1):277-305, 2014. URL: http://dx.doi.org/10.1137/120880240.
  2. Marek Cygan, Fedor V. Fomin, Lukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michal Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 2015. URL: http://dx.doi.org/10.1007/978-3-319-21275-3.
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  5. Rodney G. Downey and Michael R. Fellows. Fundamentals of Parameterized Complexity. Texts in Computer Science. Springer, 2013. URL: http://dx.doi.org/10.1007/978-1-4471-5559-1.
  6. Jirí Fiala, Petr A. Golovach, and Jan Kratochvíl. Parameterized complexity of coloring problems: Treewidth versus vertex cover. Theor. Comput. Sci., 412(23):2513-2523, 2011. URL: http://dx.doi.org/10.1016/j.tcs.2010.10.043.
  7. Lars Jaffke and Bart M. P. Jansen. Fine-grained parameterized complexity analysis of graph coloring problems. In Dimitris Fotakis, Aris Pagourtzis, and Vangelis Th. Paschos, editors, Algorithms and Complexity - 10th International Conference, CIAC 2017, Athens, Greece, May 24-26, 2017, Proceedings, volume 10236 of Lecture Notes in Computer Science, pages 345-356, 2017. URL: http://dx.doi.org/10.1007/978-3-319-57586-5_29.
  8. Bart M. P. Jansen and Stefan Kratsch. Data reduction for graph coloring problems. Inf. Comput., 231:70-88, 2013. URL: http://dx.doi.org/10.1016/j.ic.2013.08.005.
  9. Bart M. P. Jansen and Astrid Pieterse. Optimal sparsification for some binary csps using low-degree polynomials. In Piotr Faliszewski, Anca Muscholl, and Rolf Niedermeier, editors, 41st International Symposium on Mathematical Foundations of Computer Science, MFCS 2016, August 22-26, 2016 - Kraków, Poland, volume 58 of LIPIcs, pages 71:1-71:14. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2016. URL: http://dx.doi.org/10.4230/LIPIcs.MFCS.2016.71.
  10. Bart M. P. Jansen and Astrid Pieterse. Sparsification upper and lower bounds for graph problems and not-all-equal SAT. Algorithmica, 79(1):3-28, 2017. URL: http://dx.doi.org/10.1007/s00453-016-0189-9.
  11. Stefan Kratsch, Geevarghese Philip, and Saurabh Ray. Point line cover: The easy kernel is essentially tight. ACM Trans. Algorithms, 12(3):40:1-40:16, 2016. URL: http://dx.doi.org/10.1145/2832912.