In the Weighted Treewidth-$\eta$ Deletion problem we are given a node-weighted graph $G$ and we look for a vertex subset $X$ of minimum weight such that the treewidth of $G-X$ is at most $\eta$. We show that Weighted Treewidth-$\eta$ Deletion admits a randomized polynomial-time constant-factor approximation algorithm for every fixed $\eta$. Our algorithm also works for the more general Weighted Planar $F$-M-Deletion problem. This work extends the results for unweighted graphs by [Fomin, Lokshtanov, Misra, Saurabh; FOCS '12] and answers a question posed by [Agrawal, Lokshtanov, Misra, Saurabh, Zehavi; APPROX/RANDOM '18] and [Kim, Lee, Thilikos; APPROX/RANDOM '21]. The presented algorithm is based on a novel technique of random sampling of so-called protrusions.

翻译：在带权树宽-$\eta$削减问题中，给定一个节点带权图$G$，我们需要寻找一个最小权重的顶点子集$X$，使得$G-X$的树宽不超过$\eta$。我们证明对于任意固定的$\eta$，带权树宽-$\eta$削减问题存在随机多项式时间的常数因子近似算法。该算法同样适用于更一般的带权平面$F$-M-削减问题。本研究扩展了[Fomin, Lokshtanov, Misra, Saurabh; FOCS '12]关于无权图的结果，并回答了[Agrawal, Lokshtanov, Misra, Saurabh, Zehavi; APPROX/RANDOM '18]与[Kim, Lee, Thilikos; APPROX/RANDOM '21]提出的问题。所提出的算法基于一种随机采样所谓"突起结构"的新技术。