This paper deals with the following graph partitioning problem: given a graph with n nodes, p of which are prescribed to be centers (the remaining nodes are called units), the goal is to find a partition of the set of nodes into connected components containing only one center each, so as to minimize the total assignment cost of units to centers. This problem is known to be NP-hard in general graphs, and it is shown here to remain such even if the assignment cost is monotone and the graph is bipartite. Therefore, we specialize on tree graphs to derive polynomial time algorithms. For this class of graphs we provide several reformulations of the problem as integer linear programs. Moreover, we develop a dynamic programming algorithm, whose recursion is based on solving sequences of minimum weight closure problems, that solves the problem on trees in O(n 2 p).
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