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

Authors Jungho Ahn , Eun Jung Kim , Euiwoong Lee

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ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
  • Theory of computation → Graph algorithms analysis
Keywords
  • ptolemaic
  • approximation algorithm
  • linear programming
  • feedback vertex set

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Abstract

For a family of graphs ℱ, Weighted ℱ-Deletion is the problem for which the input is a vertex weighted graph G = (V, E) and the goal is to delete S ⊆ V with minimum weight such that G⧵S ∈ 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.

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Jungho Ahn, Eun Jung Kim, and Euiwoong Lee. Towards Constant-Factor Approximation for Chordal / Distance-Hereditary Vertex Deletion. In 31st International Symposium on Algorithms and Computation (ISAAC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 181, pp. 62:1-62:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.ISAAC.2020.62

Author Details

Jungho Ahn
  • Department of Mathematical Sciences, KAIST, Daejeon, South Korea
  • Discrete Mathematics Group, Institute for Basic Science (IBS), Daejeon, South Korea
Eun Jung Kim
  • Université Paris-Dauphine, PSL University, CNRS, LAMSADE, 75016, Paris, France
Euiwoong Lee
  • Department of Computer Science, New York University, NY, USA

Funding

Jungho Ahn: the Institute for Basic Science (IBS-R029-C1). Eun Jung Kim: ANR JCJC project "ASSK" (ANR-18-CE40-0025-01). Euiwong Lee: Simons Collaboration on Algorithms and Geometry.

Acknowledgements

A part of this research was done during the "2019 IBS Summer Research Program on Algorithms and Complexity in Discrete Structures", hosted by the IBS Discrete Mathematics Group.

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