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Parameterized Study of Steiner Tree on Unit Disk Graphs

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Abstract

We study the Steiner Tree problem on unit disk graphs. Given a n vertex unit disk graph G, a subset \(R\subseteq V(G)\) 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 \(V\setminus R\). The vertices of R are referred to as terminals and the vertices of \(V(G)\setminus R\) 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 \(n^{O(\sqrt{t+k})}\) 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 (Discret. Comput. Geometry 62(4):879–911, 2019). We mention that the algorithmic results can be made to work for Steiner Tree problem on disk graphs with bounded aspect ratio. Finally, we prove that Steiner Tree problem on disk graphs parameterized by k, is W[1]-hard.

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Notes

  1. For any \(\epsilon >0\) computes a \((1+\epsilon )\) approximation in time \(f(p,\epsilon )\times n^{O(1)}\) for a computable function f independent of n.

  2. A full Steiner tree is a Steiner tree which has all the terminal vertices as its leaves.

  3. \(k\times k\) Grid Tiling with \(\ge \) problem is W[1]-hard, assuming ETH, cannot be solved in \(f(k)n^{o(k)}\) for any function f.

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Acknowledgements

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

Funding

Research of Sujoy Bhore was supported by the Austrian Science Fund (FWF), grant P 31119. 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. 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.

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Authors and Affiliations

  1. Indian Institute of Science Education and Research, Bhopal, India

    Sujoy Bhore

  2. Ben-Gurion University of the Negev, Beersheba, Israel

    Paz Carmi & Meirav Zehavi

  3. Indian Institute of Technology, Kharagpur, India

    Sudeshna Kolay

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  1. Sujoy Bhore
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  2. Paz Carmi
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Correspondence to Sujoy Bhore.

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Bhore, S., Carmi, P., Kolay, S. et al. Parameterized Study of Steiner Tree on Unit Disk Graphs. Algorithmica 85, 133–152 (2023). https://doi.org/10.1007/s00453-022-01020-z

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