Bounded Size Graph Clustering on Trees

Given an undirected graph with vertex and edge weights and a subset of vertices called terminals, the bounded size graph clustering (BSGC) problem is to partition the vertices into clusters of size at most a given budget such that each cluster contains at most one terminal and the total size of the clusters is minimized, where the size of a cluster is defined as the total vertex weight in the subset plus the total edge weight at the boundary of the cluster. For BSGC on trees, we present a pseudo-polynomial time exact algorithm, which can be modified via rounding to yield a (1+ε)-approximation in polynomial time violating the given budget by a 1+ε factor.