Kawarabayashi and Sidiropoulos [KS22] obtained an $O(\log^2 n)$-approximation algorithm for Multicut in planar digraphs via a natural LP relaxation, which also establishes a corresponding upper bound on the multicommodity flow-cut gap. Their result is in contrast to a lower bound of $\tildeΩ(n^{1/7})$ on the flow-cut gap for general digraphs due to Chuzhoy and Khanna [CK09]. We extend the algorithm and analysis in [KS22] to the node-weighted Multicut problem. Unlike in general digraphs, node-weighted problems cannot be reduced to edge-weighted problems in a black box fashion due to the planarity restriction. We use the node-weighted problem as a vehicle to accomplish two additional goals: (i) to obtain a deterministic algorithm (the algorithm in [KS22] is randomized), and (ii) to simplify and clarify some aspects of the algorithm and analysis from [KS22]. The Multicut result, via a standard technique, implies an approximation for the Nonuniform Sparsest Cut problem with an additional logarithmic factor loss.

翻译：Kawarabayashi 与 Sidiropoulos [KS22] 通过一个自然的线性规划松弛，为平面有向图中的多割问题获得了一个 $O(\log^2 n)$ 近似算法，同时也为多商品流-割间隙建立了一个相应的上界。他们的结果与 Chuzhoy 和 Khanna [CK09] 针对一般有向图所证明的 $\tildeΩ(n^{1/7})$ 流-割间隙下界形成了对比。我们将 [KS22] 中的算法与分析扩展到了节点加权的多割问题。与一般有向图不同，由于平面性限制，节点加权问题无法以黑盒方式归约到边加权问题。我们以节点加权问题为载体，实现了两个额外目标：(i) 获得一个确定性算法（[KS22] 中的算法是随机的），以及 (ii) 简化和澄清 [KS22] 中算法与分析的某些方面。通过一项标准技术，该多割结果意味着对非均匀稀疏割问题也能得到一个近似解，但需额外付出一个对数因子的代价。