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Parameterized Study of Steiner Tree on Unit Disk Graphs

Authors Sujoy Bhore , Paz Carmi , Sudeshna Kolay , Meirav Zehavi

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Author Details

Sujoy Bhore
  • Algorithms and Complexity Group, TU Wien, Austria
Paz Carmi
  • Ben-Gurion University of the Negev, Beersheba, Israel
Sudeshna Kolay
  • Indian Institute of Technology Kharagpur, India
Meirav Zehavi
  • Ben-Gurion University of the Negev, Beersheba, Israel

Funding

  • Bhore, Sujoy: Research of Sujoy Bhore was supported by the Austrian Science Fund (FWF), grant P 31119.
  • Carmi, Paz: Research of Paz Carmi was partially supported by the Lynn and William Frankel Center for Computer Science and by Grant 2016116 from the United States-Israel Binational Science Foundation.
  • Zehavi, Meirav: Research of Meirav Zehavi was supported by the Israel Science Foundation (ISF) grant no. 1176/18 and United States - Israel Binational Science Foundation (BSF) grant no. 2018302.

Acknowledgements

We are grateful to the anonymous reviewers for their helpful comments.

Cite As Get BibTex

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. Google Scholar
  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. Google Scholar
  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. Google Scholar
  4. Piotr Berman and Viswanathan Ramaiyer. Improved approximations for the steiner tree problem. Journal of Algorithms, 17(3):381-408, 1994. Google Scholar
  5. Ahmad Biniaz, Anil Maheshwari, and Michiel Smid. On full steiner trees in unit disk graphs. Computational Geometry, 48(6):453-458, 2015. Google Scholar
  6. Hans L. Bodlaender. A linear time algorithm for finding tree-decompositions of small treewidth, 1996. Google Scholar
  7. Al Borchers and Ding-Zhu Du. Thek-steiner ratio in graphs. SIAM Journal on Computing, 26(3):857-869, 1997. Google Scholar
  8. Janka Chlebikova and M Chlebík. The steiner tree problem on graphs: Inapproximability results. Theoretical Computer Science, 406(3):207-214, 2008. Google Scholar
  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. Google Scholar
  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. Google Scholar
  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. Google Scholar
  12. Rodney G Downey and Michael Ralph Fellows. Parameterized complexity. Springer Science & Business Media, 2012. Google Scholar
  13. Stuart E Dreyfus and Robert A Wagner. The steiner problem in graphs. Networks, 1(3):195-207, 1971. Google Scholar
  14. Adrian Dumitrescu and János Pach. Minimum clique partition in unit disk graphs. In Graphs and Combinatorics, volume 27, pages 399-411. Springer, 2011. Google Scholar
  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. Google Scholar
  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. Google Scholar
  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. Google Scholar
  19. William K Hale. Frequency assignment: Theory and applications. Proceedings of the IEEE, 68(12):1497-1514, 1980. Google Scholar
  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. Google Scholar
  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. Google Scholar
  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. Google Scholar
  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. Google Scholar
  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. Google Scholar
  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. Google Scholar
  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. Google Scholar
  29. Ondřej Suchy. Extending the kernel for planar steiner tree to the number of steiner vertices. Algorithmica, 79(1):189-210, 2017. Google Scholar
  30. Vijay V Vazirani. Approximation algorithms. Springer Science & Business Media, 2013. Google Scholar
  31. DW Wang and Yue-Sun Kuo. A study on two geometric location problems. Information processing letters, 28(6):281-286, 1988. Google Scholar
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