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Approximate association via dissociation. (English) Zbl 1354.05110
Summary: A vertex set $$X$$ of a graph $$G$$ is an association set if each component of $$G-X$$ is a clique, and a dissociation set if each of these cliques has only one or two vertices. We observe some special structures and show that if none of them exists, then the minimum association set problem can be reduced to the minimum weighted dissociation set problem. This yields the first nontrivial approximation algorithm for the association set problem, with approximation ratio 2.5. The reduction is based on a combinatorial study of modular decomposition of graphs free of these special structures. Further, a novel algorithmic use of modular decomposition enables us to implement our algorithm in $$O(m n + n^2)$$ time.

##### MSC:
 05C85 Graph algorithms (graph-theoretic aspects) 05C75 Structural characterization of families of graphs 68R10 Graph theory (including graph drawing) in computer science 68W25 Approximation algorithms
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