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# On the Parameterized Complexity of Grid Contraction

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LIPIcs.SWAT.2020.34.pdf
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## Acknowledgements

Some part of this project was completed when the third author was a Senior Research Fellow at The Institute of Mathematical Sciences, HBNI, Chennai, India. The project started when the second and the third authors were visiting Prof. Michael Fellows at the University of Bergen, Bergen, Norway. Both the authors would like to thank Prof. Michael Fellows for the invitation.

## Cite As

Saket Saurabh, Uéverton dos Santos Souza, and Prafullkumar Tale. On the Parameterized Complexity of Grid Contraction. In 17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 162, pp. 34:1-34:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.SWAT.2020.34

## Abstract

For a family of graphs 𝒢, the 𝒢-Contraction problem takes as an input a graph G and an integer k, and the goal is to decide if there exists F ⊆ E(G) of size at most k such that G/F belongs to 𝒢. Here, G/F is the graph obtained from G by contracting all the edges in F. In this article, we initiate the study of Grid Contraction from the parameterized complexity point of view. We present a fixed parameter tractable algorithm, running in time c^k ⋅ |V(G)|^{{O}(1)}, for this problem. We complement this result by proving that unless ETH fails, there is no algorithm for Grid Contraction with running time c^{o(k)} ⋅ |V(G)|^{{O}(1)}. We also present a polynomial kernel for this problem.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Fixed parameter tractability
##### Keywords
• Grid Contraction
• FPT
• Kernelization
• Lower Bound

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## References

1. Akanksha Agrawal, Fedor Fomin, Daniel Lokshtanov, Saket Saurabh, and Prafullkumar Tale. Path contraction faster than 2ⁿ. The 46th International Colloquium on Automata, Languages and Programming (ICALP 2019), 2019.
2. Akanksha Agrawal, Daniel Lokshtanov, Saket Saurabh, and Meirav Zehavi. Split contraction: The untold story. ACM Transactions on Computation Theory (TOCT), 11(3):1-22, 2019.
3. Takao Asano and Tomio Hirata. Edge-Contraction Problems. Journal of Computer and System Sciences, 26(2):197-208, 1983.
4. Rémy Belmonte, Petr A. Golovach, Pim Hof, and Daniël Paulusma. Parameterized complexity of three edge contraction problems with degree constraints. Acta Informatica, 51(7):473-497, 2014.
5. Andries Evert Brouwer and Henk Jan Veldman. Contractibility and NP-completeness. Journal of Graph Theory, 11(1):71-79, 1987.
6. Leizhen Cai and Chengwei Guo. Contracting few edges to remove forbidden induced subgraphs. In International Symposium on Parameterized and Exact Computation, pages 97-109. Springer, 2013.
7. Marek Cygan, Fedor V. Fomin, Lukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michal Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 2015.
8. Rod G. Downey and Michael R. Fellows. Fundamentals of Parameterized complexity. Springer-Verlag, 2013.
9. Jörg Flum and Martin Grohe. Parameterized Complexity Theory. Texts in Theoretical Computer Science. An EATCS Series. Springer, 2006.
10. Fedor V Fomin, Daniel Lokshtanov, Saket Saurabh, and Meirav Zehavi. Kernelization: theory of parameterized preprocessing. Cambridge University Press, 2019.
11. Petr A Golovach, Marcin Kamiński, Daniël Paulusma, and Dimitrios M Thilikos. Increasing the minimum degree of a graph by contractions. Theoretical computer science, 481:74-84, 2013.
12. Petr A. Golovach, Pim van 't Hof, and Daniel Paulusma. Obtaining planarity by contracting few edges. Theoretical Computer Science, 476:38-46, 2013.
13. Sylvain Guillemot and Dániel Marx. A faster FPT algorithm for bipartite contraction. Inf. Process. Lett., 113(22-24):906-912, 2013.
14. Pinar Heggernes, Pim van 't Hof, Daniel Lokshtanov, and Christophe Paul. Obtaining a bipartite graph by contracting few edges. SIAM Journal on Discrete Mathematics, 27(4):2143-2156, 2013.
15. Pinar Heggernes, Pim Van’t Hof, Benjamin Lévêque, Daniel Lokshtanov, and Christophe Paul. Contracting graphs to paths and trees. Algorithmica, 68(1):109-132, 2014.
16. R. Krithika, Pranabendu Misra, Ashutosh Rai, and Prafullkumar Tale. Lossy kernels for graph contraction problems. In 36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science, FSTTCS 2016, pages 23:1-23:14, 2016.
17. R Krithika, Pranabendu Misra, and Prafullkumar Tale. An FPT algorithm for contraction to cactus. In International Computing and Combinatorics Conference, pages 341-352. Springer, 2018.
18. Daniel Lokshtanov, Neeldhara Misra, and Saket Saurabh. On the hardness of eliminating small induced subgraphs by contracting edges. In International Symposium on Parameterized and Exact Computation, pages 243-254, 2013.
19. Rolf Niedermeier. Invitation to fixed-parameter algorithms. Oxford Lecture Series in Mathematics and Its Applications. Oxford University Press, 2006.
20. Toshimasa Watanabe, Tadashi Ae, and Akira Nakamura. On the removal of forbidden graphs by edge-deletion or by edge-contraction. Discrete Applied Mathematics, 3(2):151-153, 1981.
21. Toshimasa Watanabe, Tadashi Ae, and Akira Nakamura. On the NP-hardness of edge-deletion and-contraction problems. Discrete Applied Mathematics, 6(1):63-78, 1983.
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