We present an $O(\log k)$-approximation for both the edge-weighted and node-weighted versions of \DST in planar graphs where $k$ is the number of terminals. We extend our approach to \MDST (in general graphs \MDST and \DST are easily seen to be equivalent but in planar graphs this is not the case necessarily) in which we get a $O(R+\log k)$-approximation for planar graphs for where $R$ is the number of roots.

翻译：我们针对平面图中边加权和节点加权两种版本的\textsc{有向斯坦纳树}（\DST）问题，给出$O(\log k)$-近似算法，其中$k$为终端数量。我们将该方法推广至\textsc{多根有向斯坦纳树}（\MDST）问题——注意在一般图中\MDST 与\DST 显然等价，但在平面图中这一等价性未必成立——并得到平面图上$O(R+\log k)$-近似算法，其中$R$为根节点数量。