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

TL;DR

This paper develops a constant-factor approximation algorithm for the Weighted F-Deletion problem where F is the intersection of chordal and distance-hereditary graphs, expanding efficient solutions to a new class of perfect graphs.

Contribution

It introduces the first constant-factor approximation algorithm for F-Deletion when F is ptolemaic graphs, a class combining chordal and distance-hereditary properties.

Findings

Provides a new approximation algorithm for ptolemaic graphs.

Establishes properties of inter-clique digraphs relevant to the problem.

Develops an approximation for Feedback Vertex Set with Precedence Constraints.

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 $F$ 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.