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The paper considers a generalization of the optimum requirement spanning tree problem (ORST problem) ®rst studied by Hu in 1974. Originally, ORST was regarded as a communication network of tree type with the minimum average cost, and it is obtained by the well-known Gomory±Hu algorithm when the degrees of vertices are not restricted. The ORST problem is generalized by (i) generalizing the objective function and (ii) imposing maximum degree constraints. The generalized ORST problem includes some practical problems, one of which is proposed in this paper, but is not ef®ciently solvable in general. However, I show that a particular tree (which is obtained by a sort of greedy algorithm but is explicitly de®nable) is a solution of the generalized problem when a certain practical condition is satis®ed. The condition is closely related to the Monge property, which is originally discussed in the Hitchcock transportation problem, and is known to make some NP-hard problems ef®ciently solvable.

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In this paper, we propose three O(noS(m,n)