Towards constant-factor approximation for chordal / distance-hereditary vertex deletion

Abstract

For a family of graphs F, Weighted F-Deletion is the problem for which the input is a vertex weighted graph G " pV, Eq and the goal is to delete S Ď V with minimum weight such that GzS P F. Designing a constant-factor approximation algorithm for large subclasses of perfect graphs has been an interesting research direction. Block graphs, 3leaf power graphs, and interval graphs are known to admit constant-factor approximation algorithms, but the question is open for chordal graphs and distance-hereditary graphs.In this paper, we add one more class to this list by presenting a constant-factor approximation algorithm when F is the intersection of chordal graphs and distancehereditary graphs. They are known as ptolemaic graphs and form a superset of both block graphs and 3-leaf power graphs above. Our proof presents new properties and algorithmic results on inter-clique digraphs as well as an approximation algorithm for a variant of Feedback Vertex Set that exploits this relationship (named Feedback Vertex Set with Precedence Constraints), each of which may be of independent interest.