Clustering with Local Restrictions
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We study a family of graph clustering problems where each cluster has to satisfy a certain local requirement. Formally, let μ be a function on the subsets of vertices of a graph G. In the (μ,p,q)-PARTITION problem, the task is to find a partition of the vertices into clusters where each cluster C satisfies the requirements that (1) at most q edges leave C and (2) μ(C)< p. Our first result shows that if μ is an arbitrary polynomial-time computable monotone function, then (μ,p,q)-PARTITION can be solved in time n^O(q), i.e., it is polynomial-time solvable for every fixed q. We study in detail three concrete functions μ (the number of vertices in the cluster, number of nonedges in the cluster, maximum number of non-neighbors a vertex has in the cluster), which correspond to natural clustering problems. For these functions, we show that (μ,p,q)-PARTITION can be solved in time 2^O(p)· n^O(1) and in time 2^O(q)· n^O(1) on n-vertex graphs, i.e., the problem is fixed-parameter tractable parameterized by p or by q.
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