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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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This study is motivated by an electoral application where we look into the following question: how much biased can the assignment of parliament seats be in a majority system under the effect of vicious gerrymandering when the two competing parties have the same electoral strength? To give a first theoretical answer to this question, we introduce a stylized combinatorial model, where the territory is represented by a rectangular grid graph, the vote outcome by a "balanced" red/blue node bicoloring and a district map by a connected partition of the grid whose components all have the same size. We constructively prove the existence in cycles and grid graphs of a balanced bicoloring and of two antagonist "partisan" district maps such that the discrepancy between their number of "red" (or "blue") districts for that bicoloring is extremely large, in fact as large as allowed by color balance. © 2009 Elsevier B.V. All rights reserved