  A vertex set $X$ of a graph $G$ is an association set if each component of $G
- X$ is a clique, or a dissociation set if each component of $G - X$ is a
single vertex or a single edge. Interestingly, $G - X$ is then precisely a
graph containing no induced $P_3$'s or containing no $P_3$'s, respectively. 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 association set, and its approximation ratio is 2.5, matching the
best result of the closely related cluster editing problem. 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 this approach in $O(m n + n^2)$ time.
