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Parameterized Pre-Coloring Extension and List Coloring Problems

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Gregory Gutin, Diptapriyo Majumdar, Sebastian Ordyniak, and Magnus Wahlström. Parameterized Pre-Coloring Extension and List Coloring Problems. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 19:1-19:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.STACS.2020.19

Abstract

Golovach, Paulusma and Song (Inf. Comput. 2014) asked to determine the parameterized complexity of the following problems parameterized by k: (1) Given a graph G, a clique modulator D (a clique modulator is a set of vertices, whose removal results in a clique) of size k for G, and a list L(v) of colors for every v ∈ V(G), decide whether G has a proper list coloring; (2) Given a graph G, a clique modulator D of size k for G, and a pre-coloring λ_P: X → Q for X ⊆ V(G), decide whether λ_P can be extended to a proper coloring of G using only colors from Q. For Problem 1 we design an O*(2^k)-time randomized algorithm and for Problem 2 we obtain a kernel with at most 3k vertices. Banik et al. (IWOCA 2019) proved the following problem is fixed-parameter tractable and asked whether it admits a polynomial kernel: Given a graph G, an integer k, and a list L(v) of exactly n-k colors for every v ∈ V(G), decide whether there is a proper list coloring for G. We obtain a kernel with O(k²) vertices and colors and a compression to a variation of the problem with O(k) vertices and O(k²) colors.

Subject Classification

ACM Subject Classification
• Theory of computation → Parameterized complexity and exact algorithms
Keywords
• Parameterized Algorithms
• W-hardness
• Kernelization
• Graph Coloring
• List Coloring

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References

1. Noga Alon, Gregory Z. Gutin, Eun Jung Kim, Stefan Szeider, and Anders Yeo. Solving MAX-r-SAT above a tight lower bound. Algorithmica, 61(3):638-655, 2011.
2. Pranav Arora, Aritra Banik, Vijay Kumar Paliwal, and Venkatesh Raman. Some (in)tractable parameterizations of coloring and list-coloring. In Jianer Chen and Pinyan Lu, editors, Frontiers in Algorithmics - 12th International Workshop, FAW 2018, Proceedings, volume 10823 of Lecture Notes in Computer Science, pages 126-139. Springer, 2018.
3. Aritra Banik, Ashwin Jacob, Vijay Kumar Paliwal, and Venkatesh Raman. Fixed-parameter tractability of (n-k) List Coloring. In C.J. Colbourn, R. Grossi, and N. Pisani, editors, IWOCA 2019, Proceedings, volume 11638 of Lecture Notes in Computer Science, pages 61-69, 2019. URL: https://doi.org/10.1007/978-3-030-25005-8_6.
4. D. Berend and T. Tassa. Improved bounds on bell numbers and on moments of sums of random variables. Probability and Mathematical Statistics, 30:185-205, 2010.
5. Andreas Björklund, Thore Husfeldt, and Mikko Koivisto. Set partitioning via inclusion-exclusion. SIAM J. Comput., 39(2):546-563, 2009.
6. Hans L. Bodlaender, Bart M. P. Jansen, and Stefan Kratsch. Kernel bounds for path and cycle problems. Theor. Comput. Sci., 511:117-136, 2013.
7. James R. Bunch and John E. Hopcroft. Triangular factorization and inversion by fast matrix multiplication. Mathematics of Computation, 28(125):231-236, 1974.
8. Leizhen Cai. Parameterized complexity of vertex colouring. Discrete Applied Mathematics, 127(3):415-429, 2003.
9. Benny Chor, Mike Fellows, and David Juedes. Linear kernels in linear time, or how to save k colors in O(n²) steps. In Proceedings of the 30th International Conference on Graph-Theoretic Concepts in Computer Science, WG'04, pages 257-269. Springer-Verlag, 2004.
10. Maria Chudnovsky. Coloring graphs with forbidden induced subgraphs. In 2014 International Congress of Mathematicians, volume IV, pages 291-302, 2014.
11. M. Cygan, F.V. Fomin, L. Kowalik, D. Lokshtanov, D. Marx, M. Pilipczuk, M. Pilipczuk, and S. Saurabh. Parameterized Algorithms. Springer, 2015.
12. Marek Cygan, Holger Dell, Daniel Lokshtanov, Dániel Marx, Jesper Nederlof, Yoshio Okamoto, Ramamohan Paturi, Saket Saurabh, and Magnus Wahlström. On problems as hard as CNF-SAT. ACM Trans. Algorithms, 12(3):41:1-41:24, 2016.
13. Reinhard Diestel. Graph Theory, 4th Edition, volume 173 of Graduate texts in mathematics. Springer, 2012.
14. Michael Dom, Daniel Lokshtanov, and Saket Saurabh. Kernelization Lower Bounds Through Colors and IDs. ACM Trans. Algorithms, 11(2):13:1-13:20, 2014.
15. R.G. Downey and M.R. Fellows. Fundamentals of Parameterized Complexity. Springer, 2013.
16. Bruno Escoffier. Saving colors and max coloring: Some fixed-parameter tractability results. Theor. Comput. Sci., 758:30-41, 2019. URL: https://doi.org/10.1016/j.tcs.2018.08.002.
17. F.V. Fomin, D. Lokshtanov, S. Saurabh, and M. Zehavi. Kernelization. Cambridge Univ. Press, 2019.
18. Petr A. Golovach, Matthew Johnson, Daniël Paulusma, and Jian Song. A survey on the computational complexity of coloring graphs with forbidden subgraphs. Journal of Graph Theory, 84(4):331-363, 2017.
19. Petr A. Golovach, Daniël Paulusma, and Jian Song. Closing complexity gaps for coloring problems on h-free graphs. Inf. Comput., 237:204-214, 2014.
20. 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.
21. Tommy R. Jensen and Bjarne Toft. Graph Coloring Problems. Wiley & Sons, 1994.
22. Robert Krauthgamer and Ohad Trabelsi. The set cover conjecture and subgraph isomorphism with a tree pattern. In Rolf Niedermeier and Christophe Paul, editors, 36th International Symposium on Theoretical Aspects of Computer Science, STACS 2019, March 13-16, 2019, Berlin, Germany, volume 126 of LIPIcs, pages 45:1-45:15. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2019. URL: https://doi.org/10.4230/LIPIcs.STACS.2019.45.
23. L. Lovász. Coverings and coloring of hypergraphs. In 4th Southeastern Conference on Combinatorics, Graph Theory, and Computing, Utilitas Math, pages 3-12, 1973.
24. Daniël Paulusma. Open problems on graph coloring for special graph classes. In Ernst W. Mayr, editor, Graph-Theoretic Concepts in Computer Science - 41st International Workshop, WG 2015, Garching, Germany, June 17-19, 2015, Revised Papers, volume 9224 of Lecture Notes in Computer Science, pages 16-30. Springer, 2015.
25. B. Randerath and I. Schiermeyer. Vertex coloring and forbidden subgraphs - a survey. Graphs Comb., 20:1-40, 2004.
26. J. T. Schwartz. Fast probabilistic algorithms for verification of polynomial identities. J. ACM, 27(4):701-717, October 1980.
27. Z. Tuza. Graph colorings with local restrictions - a survey. Discussiones Mathematicae Graph Theory, 17:161-228, 1997.
28. Magnus Wahlström. Abusing the Tutte matrix: An algebraic instance compression for the K-set-cycle problem. In Natacha Portier and Thomas Wilke, editors, 30th International Symposium on Theoretical Aspects of Computer Science, STACS 2013, February 27 - March 2, 2013, Kiel, Germany, volume 20 of LIPIcs, pages 341-352. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2013.
29. F. Yates. The design and analysis of factorial experiments. Technical Report Technical Communication no. 35, Commonwealth Bureau of Soils, 1937.
30. Richard Zippel. Probabilistic algorithms for sparse polynomials. In Proceedings of the International Symposiumon on Symbolic and Algebraic Computation, EUROSAM '79, pages 216-226, 1979.
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