Towards Constant-Factor Approximation for Chordal/Distance-Hereditary Vertex Deletion

Abstract

For a family of graphs \(\mathcal {F}\), Weighted \(\mathcal {F}\)-Deletion is the problem for which the input is a vertex weighted graph \(G = (V, E)\) and the goal is to delete \(S \subseteq V\) with minimum weight such that \(G \setminus S \in \mathcal {F}\). Designing a constant-factor approximation algorithm for large subclasses of perfect graphs has been an interesting research direction. Block graphs, 3-leaf 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 \(\mathcal {F}\) is the intersection of chordal graphs and distance-hereditary 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.