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# Parameterized Study of Steiner Tree on Unit Disk Graphs

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LIPIcs.SWAT.2020.13.pdf
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## Cite As

Sujoy Bhore, Paz Carmi, Sudeshna Kolay, and Meirav Zehavi. Parameterized Study of Steiner Tree on Unit Disk Graphs. In 17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 162, pp. 13:1-13:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.SWAT.2020.13

## Abstract

We study the Steiner Tree problem on unit disk graphs. Given a n vertex unit disk graph G, a subset R⊆ V(G) of t vertices and a positive integer k, the objective is to decide if there exists a tree T in G that spans over all vertices of R and uses at most k vertices from V⧵ R. The vertices of R are referred to as terminals and the vertices of V(G)⧵ R as Steiner vertices. First, we show that the problem is NP-hard. Next, we prove that the Steiner Tree problem on unit disk graphs can be solved in n^{O(√{t+k})} time. We also show that the Steiner Tree problem on unit disk graphs parameterized by k has an FPT algorithm with running time 2^{O(k)}n^{O(1)}. In fact, the algorithms are designed for a more general class of graphs, called clique-grid graphs [Fomin et al., 2019]. We mention that the algorithmic results can be made to work for Steiner Tree on disk graphs with bounded aspect ratio. Finally, we prove that Steiner Tree on disk graphs parameterized by k is W[1]-hard.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Computational geometry
##### Keywords
• Unit Disk Graphs
• FPT
• Subexponential exact algorithms
• NP-Hardness
• W-Hardness

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## References

1. A Karim Abu-Affash. The euclidean bottleneck full steiner tree problem. Algorithmica, 71(1):139-151, 2015.
2. Sanjeev Arora. Polynomial time approximation schemes for euclidean traveling salesman and other geometric problems. Journal of the ACM (JACM), 45(5):753-782, 1998.
3. Piotr Berman, Marek Karpinski, and Alexander Zelikovsky. 1.25-approximation algorithm for steiner tree problem with distances 1 and 2. In Workshop on Algorithms and Data Structures, pages 86-97. Springer, 2009.
4. Piotr Berman and Viswanathan Ramaiyer. Improved approximations for the steiner tree problem. Journal of Algorithms, 17(3):381-408, 1994.
5. Ahmad Biniaz, Anil Maheshwari, and Michiel Smid. On full steiner trees in unit disk graphs. Computational Geometry, 48(6):453-458, 2015.
6. Hans L. Bodlaender. A linear time algorithm for finding tree-decompositions of small treewidth, 1996.
7. Al Borchers and Ding-Zhu Du. Thek-steiner ratio in graphs. SIAM Journal on Computing, 26(3):857-869, 1997.
8. Janka Chlebikova and M Chlebík. The steiner tree problem on graphs: Inapproximability results. Theoretical Computer Science, 406(3):207-214, 2008.
9. Brent N Clark, Charles J Colbourn, and David S Johnson. Unit disk graphs. In Annals of Discrete Mathematics, volume 48, pages 165-177. Elsevier, 1991.
10. Marek Cygan, Fedor V Fomin, Łukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michał Pilipczuk, and Saket Saurabh. Parameterized algorithms, volume 4. Springer, 2015.
11. Mark de Berg, Hans L Bodlaender, Sándor Kisfaludi-Bak, Dániel Marx, and Tom C van der Zanden. A framework for eth-tight algorithms and lower bounds in geometric intersection graphs. In Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, pages 574-586, 2018.
12. Rodney G Downey and Michael Ralph Fellows. Parameterized complexity. Springer Science & Business Media, 2012.
13. Stuart E Dreyfus and Robert A Wagner. The steiner problem in graphs. Networks, 1(3):195-207, 1971.
14. Adrian Dumitrescu and János Pach. Minimum clique partition in unit disk graphs. In Graphs and Combinatorics, volume 27, pages 399-411. Springer, 2011.
15. Pavel Dvořák, Andreas Emil Feldmann, Dušan Knop, Tomáš Masařík, Tomáš Toufar, and Pavel Vesely. Parameterized approximation schemes for steiner trees with small number of steiner vertices. arXiv preprint arXiv:1710.00668, 2017.
16. Fedor V Fomin, Daniel Lokshtanov, Fahad Panolan, Saket Saurabh, and Meirav Zehavi. Finding, hitting and packing cycles in subexponential time on unit disk graphs. Discrete & Computational Geometry, 62(4):879-911, 2019.
17. Bernhard Fuchs, Walter Kern, Daniel Mölle, Stefan Richter, Peter Rossmanith, and Xinhui Wang. Dynamic programming for minimum Steiner trees. Theory of Computing System, 41(3):493-500, 2007. URL: https://doi.org/10.1007/s00224-007-1324-4.
18. Michael R Garey and David S. Johnson. The rectilinear steiner tree problem is np-complete. SIAM Journal on Applied Mathematics, 32(4):826-834, 1977.
19. William K Hale. Frequency assignment: Theory and applications. Proceedings of the IEEE, 68(12):1497-1514, 1980.
20. Mark Jones, Daniel Lokshtanov, MS Ramanujan, Saket Saurabh, and Ondřej Suchy. Parameterized complexity of directed steiner tree on sparse graphs. In European Symposium on Algorithms, pages 671-682. Springer, 2013.
21. Karl Kammerlander. C 900-an advanced mobile radio telephone system with optimum frequency utilization. IEEE journal on selected areas in communications, 2(4):589-597, 1984.
22. Richard M. Karp. Reducibility among combinatorial problems. In R. E. Miller, J. W. Thatcher, and J. D. Bohlinger, editors, Complexity of Computer Computations, pages 85-103, 1972. URL: https://doi.org/10.1007/978-1-4684-2001-2_9.
23. Marek Karpinski and Alexander Zelikovsky. New approximation algorithms for the steiner tree problems. Journal of Combinatorial Optimization, 1(1):47-65, 1997.
24. Xianyue Li, Xiao-Hua Xu, Feng Zou, Hongwei Du, Pengjun Wan, Yuexuan Wang, and Weili Wu. A ptas for node-weighted steiner tree in unit disk graphs. In International Conference on Combinatorial Optimization and Applications, pages 36-48. Springer, 2009.
25. Dániel Marx, Marcin Pilipczuk, and Michał Pilipczuk. On subexponential parameterized algorithms for steiner tree and directed subset tsp on planar graphs. In 2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS), pages 474-484. IEEE, 2018.
26. Jesper Nederlof. Fast polynomial-space algorithms using inclusion-exclusion. Algorithmica, 65(4):868-884, 2013. URL: https://doi.org/10.1007/s00453-012-9630-x.
27. Hans Jürgen Prömel and Angelika Steger. A new approximation algorithm for the steiner tree problem with performance ratio 5/3. Journal of Algorithms, 36(1):89-101, 2000.
28. Walter Schnyder. Embedding planar graphs on the grid. In Proceedings of the first annual ACM-SIAM symposium on Discrete algorithms, pages 138-148, 1990.
29. Ondřej Suchy. Extending the kernel for planar steiner tree to the number of steiner vertices. Algorithmica, 79(1):189-210, 2017.
30. Vijay V Vazirani. Approximation algorithms. Springer Science & Business Media, 2013.
31. DW Wang and Yue-Sun Kuo. A study on two geometric location problems. Information processing letters, 28(6):281-286, 1988.
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