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

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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.

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Notes

  1. Let \(C_{\geqslant k}\) be the set of cycles of length at least k.

  2. Any graph has either a vertex-disjoint packing of \(k+1\) cycles, or a feedback vertex set of size \(O(k\log k)\).

  3. The name ptolemaic comes from the fact that the shortest path distance satisfies Ptolemy’s inequality: For every four vertices u, v, w, x, the inequality \(d(u,v)d(w,x)+d(u,x)d(v,w)\geqslant d(u,w)d(v,x)\) holds.

  4. [10] added an additional cycle covering constraint in the LP. We find it conceptually easier to deal with the last remaining cycle separately at the end.

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Authors and Affiliations

  1. Department of Mathematical Sciences, KAIST, Daejeon, South Korea

    Jungho Ahn

  2. Discrete Mathematics Group, Institute for Basic Science (IBS), Daejeon, South Korea

    Jungho Ahn

  3. Université Paris-Dauphine, PSL University, CNRS, LAMSADE, 75016, Paris, France

    Eun Jung Kim

  4. Department of EECS, University of Michigan, Ann Arbor, USA

    Euiwoong Lee

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Jungho Ahn is supported by the Institute for Basic Science (IBS-R029-C1). Eun Jung Kim is supported by the Grant from French National Research Agency (ANR) under JCJC program (ASSK: ANR-18-CE40-0025-01). Euiwoong Lee is supported by Simons Collaboration on Algorithms and Geometry.

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Ahn, J., Kim, E.J. & Lee, E. Towards Constant-Factor Approximation for Chordal/Distance-Hereditary Vertex Deletion. Algorithmica 84, 2106–2133 (2022). https://doi.org/10.1007/s00453-022-00963-7

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  • DOI: https://doi.org/10.1007/s00453-022-00963-7

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