Polylogarithmic approximation algorithms for weighted-F-Deletion problems

Let F be a family of graphs. A canonical vertex deletion problem corresponding to F 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. This is known as Weighted F Vertex Deletion problem. In this paper we devise a recursive scheme to obtain O(log^O(1) n)-A pproximation algorithms for such problems, building upon the classical technique of finding balanced separators in a graph. Roughly speaking, our scheme applies to those problems, where an optimum solution S together with a well-structured set X, form a balanced separator of the input graph. In this paper, we obtain the first O(log^O(1) n)-approximation algorithms for the following vertex deletion problems. Let F be a finite set of graphs containing a planar graph, and F = G(F) be the family of graphs such that every graph H2 G(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 (WPF-MFD) problem. We give randomized and deterministic approximation algorithms for WPF-MFD with ratios O(log^1.5 n) and O(log² n), respectively. Previously, only a randomized constant factor approximation algorithm for the unweighted version of the problem was known [FOCS 2012]. We give an O(log2 n)-factor approximation algorithm for Weighted Chordal Vertex Deletion (WCVD), 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 Weighted Distance Hereditary Vertex Deletion (WDHVD), also known as Weighted Rankwidth-1 Vertex Deletion (WR-1VD). This is the vertex deletion problem to the family of distance hereditary graphs, or equivalently, the family of graphs of rankwidth one. We believe that our recursive scheme can be applied to obtain O(logO(1) n)-approximation algorithms for many other problems as well.