A \(\boldsymbol{O}(\textbf{log}\,\boldsymbol{k})\) -Approximation for <scp>Directed Steiner Tree</scp> in Planar Graphs

We present a \(O(\log k)\) -approximation for both the edge-weighted and node-weighted versions of Directed Steiner Tree in planar graphs where \( k \) is the number of terminals. We extend our approach to Multi-Rooted Directed Steiner Tree , in which we get a \(O(R+\log k)\) -approximation for planar graphs for which \( R \) is the number of roots.