Clustering with Local Restrictions

Abstract

AbstractWe 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 (number of nonedges in the cluster, maximum number of non-neighbours a vertex has in the cluster, the number of vertices in the cluster), which correspond to natural clustering problems. For these functions, we show that (μ,p,q)-Partition can be solved in time 2O(p)·n O(1) and in randomized time 2O(q)·n O(1), i.e., the problem is fixed-parameter tractable parameterized by p or by q.KeywordsCluster ProblemLocal RestrictionParallel EdgeCorrelation ClusterSatellite ProblemThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.