Polylogarithmic Approximation Algorithms for Weighted-$\mathcal{F}$-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(logO(1) n)-approximation algorithms for such problems, building upon the classic 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(logO(1) n)approximation algorithms for the following vertex deletion problems. • We give an O(log 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(log 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 1. Our methods also allow us to obtain in a clean fashion a O(log n)-approximation algorithm for the Weighted F Vertex Deletion problem when F is a minor closed family excluding at least one planar graph. For the unweighted version of the problem constant factor approximation algorithms are were known [Fomin et al., FOCS 2012], while for the weighted version considered here an O(log n log log n)-approximation algorithm follows from [Bansal et al. SODA 2017]. We believe that our recursive scheme can be applied to obtain O(logO(1) n)-approximation algorithms for many other problems as well. ∗The research leading to these results received funding from the European Research Council under the European Unions Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreement no. 306992. †University of Bergen, Bergen, Norway. [email protected]. ‡University of Bergen, Bergen, Norway. [email protected]. §Institute of Mathematical Sciences, Chennai, India. [email protected]. ¶University of Bergen, Bergen, Norway. The Institute of Mathematical Sciences HBNI, Chennai, India. [email protected]. ‖University of Bergen, Bergen, Norway. [email protected]. ar X iv :1 70 7. 04 90 8v 1 [ cs .D S] 1 6 Ju l 2 01 7