For a family of graphs ℱ, the W<scp;>eighted</scp;> ℱ V<scp;>ertex</scp;> D<scp;>eletion</scp;> problem, is defined as follows: given an n -vertex undirected graph G and a weight function w : V ( G )࢐ ℝ, find a minimum weight subset S ⊆ V ( G ) such that G - S belongs to ℱ. We devise a recursive scheme to obtain O(log O(1) n )-approximation algorithms for such problems, building upon the classical technique of finding balanced separators . We obtain the first O(log O(1) n )-approximation algorithms for the following problems. • Let F be a finite set of graphs containing a planar graph, and ℱ= G ( F ) be the maximal family of graphs such that every graph H ∈ G ( F ) excludes all graphs in F as minors. The vertex deletion problem corresponding to ℱ= G ( F ) is the W eighted P lanar F -M inor -F ree D eletion (WP F -MFD) problem. We give a randomized and a deterministic approximation algorithms for WP F -MFD with ratios O(log 1.5 n ) and O(log 2 n ), respectively. Prior to our work, a randomized constant factor approximation algorithm for the unweighted version was known [FOCS 2012]. After our work, a deterministic constant factor approximation algorithm for the unweighted version was also obtained [SODA 2019]. • We give an O(log 2 n )-factor approximation algorithm for W eighted C hordal V ertex D eletion , the vertex deletion problem to the family of chordal graphs. On the way to this algorithm, we also obtain a constant factor approximation algorithm for M ulticut on chordal graphs. • We give an O(log 3 n )-factor approximation algorithm for W eighted D istance H ereditary V ertex D eletion . We believe that our recursive scheme can be applied to obtain O(log O(1) n )-approximation algorithms for many other problems as well.

中文翻译：

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加权-ℱ-删除问题的多对数逼近算法

对于图族 ℱ，W<scp;>eighted</scp;> ℱ V<scp;>ertex</scp;> D<scp;>eletion</scp;> 问题定义如下：一个n-顶点无向图G和权重函数w：五(G)࢐ ℝ，找到一个最小权重子集小号⊆五(G) 使得G-小号属于ℱ。我们设计了一个递归方案来获得 O(logO(1) n) - 此类问题的近似算法，建立在经典的查找技术之上平衡分离器. 我们得到第一个 O(logO(1) n) - 用于以下问题的近似算法。• 让F是包含平面图的有限图集，并且 ℱ=G(F) 是图的最大族，使得每个图H∈G(F) 排除所有图表F作为未成年人。ℱ=对应的顶点删除问题G(F) 是 W八分磷拉纳尔 F-M内-F稀土D选举（可湿性粉剂F-MFD）问题。我们给出了 WP 的随机和确定性逼近算法F-MFD，比率为 O(log1.5 n) 和 O(log2 n）， 分别。在我们的工作之前，一种随机常数因子逼近算法未加权版本已知 [FOCS 2012]。经过我们的工作，确定性常数因子逼近算法未加权还获得了版本 [SODA 2019]。• 我们给出 O(log2 n)-W 的因子逼近算法八分C大地五艾特克斯D选举, 弦图族的顶点删除问题。在实现该算法的过程中，我们还获得了 M 的常数因子逼近算法多切在弦图上。• 我们给出 O(log3 n)-W 的因子逼近算法八分D距离H世袭的五艾特克斯D选举. 我们相信我们的递归方案可以用于获得 O(logO(1) n)-近似算法也适用于许多其他问题。