Parameterized Study of Steiner Tree on Unit Disk Graphs

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.