Clustering with Local Restrictions

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, i.e., it is polynomialtime 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 ·n and in time 2 ·n on n-vertex graphs, i.e., the problem is fixed-parameter tractable parameterized by p or by q.