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.
@InProceedings{ahn_et_al:LIPIcs.ISAAC.2020.62, author = {Ahn, Jungho and Kim, Eun Jung and Lee, Euiwoong}, title = {{Towards Constant-Factor Approximation for Chordal / Distance-Hereditary Vertex Deletion}}, booktitle = {31st International Symposium on Algorithms and Computation (ISAAC 2020)}, pages = {62:1--62:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-173-3}, ISSN = {1868-8969}, year = {2020}, volume = {181}, editor = {Cao, Yixin and Cheng, Siu-Wing and Li, Minming}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2020.62}, URN = {urn:nbn:de:0030-drops-134063}, doi = {10.4230/LIPIcs.ISAAC.2020.62}, annote = {Keywords: ptolemaic, approximation algorithm, linear programming, feedback vertex set} }
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