Polylogarithmic Approximation Algorithms for Weighted-F-deletion Problems

For a family of graphs F, the Weighted F Vertex Deletion problem, is defined as follows: given an n-vertex undirected graph G and a weight function w: V(G)→ℝ F, find a minimum weight subset S¢V(G) such that G-S belongs to F. We devise a recursive scheme to obtain O(logO(1) n)-approximation algorithms for such problems, building upon the classical technique of finding balanced separators. We obtain the first O(logO(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 HϵG(F) excludes all graphs in F as minors. The vertex deletion problem corresponding to F=G(F) is the Weighted Planar F-Minor-Free Deletion (WP F-MFD) problem. We give a randomized and a deterministic approximation algorithms for WP F-MFD with ratios O(log1.5 n) and O(log2 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(log2 n)-factor approximation algorithm for Weighted Chordal Vertex Deletion, 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 Multicut on chordal graphs. We give an O(log3 n)-factor approximation algorithm for WeightedDistance Hereditary Vertex Deletion. We believe that our recursive scheme can be applied to obtain O(logO(1) n)-approximation algorithms for many other problems as well.