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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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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


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

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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)


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
  • Unit Disk Graphs
  • FPT
  • Subexponential exact algorithms
  • NP-Hardness
  • W-Hardness


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