Abstract
Given an undirected simple graph G, a set of vertices is an r-clique transversal if it has at least one vertex from every r-clique. Such sets generalize vertex covers as a vertex cover is a 2-clique transversal. Perfect graphs are a well-studied class of graphs on which a minimum weight vertex cover can be obtained in polynomial time. Further, an r-clique transversal in a perfect graph is also a set of vertices whose deletion results in an \((r-1)\)-colorable graph. In this work, we study the problem of finding a minimum weight r-clique transversal in a perfect graph. This problem is known to be \(\mathsf {NP}\)-hard for \(r \ge 3\) and admits a straightforward r-approximation algorithm. We describe two different \(\frac{r+1}{2}\)-approximation algorithms for the problem. Both the algorithms are based on (different) linear programming relaxations. The first algorithm employs the primal–dual method while the second uses rounding based on a threshold value. We also show that the problem is APX-hard and describe hardness results in the context of parameterized algorithms and kernelization.
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A preliminary version of this paper appeared in the proceedings of the 22nd European Symposium on Algorithms (ESA 2014).
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Fiorini, S., Krithika, R., Narayanaswamy, N.S. et al. Approximability of Clique Transversal in Perfect Graphs. Algorithmica 80, 2221–2239 (2018). https://doi.org/10.1007/s00453-017-0315-3
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DOI: https://doi.org/10.1007/s00453-017-0315-3