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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 AsGet 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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