Lowering eccentricity of a tree by node upgrading

The eccentricity lowering problem is to reduce the eccentricity of a communication network within a given bound, by upgrading some nodes (i.e., shrinking the lengths of the edges linking to such nodes), where we want to minimize the required cost. We consider two types of node-upgrading strategies; i.e., the continuous upgrading and the discrete upgrading, where the improvement under the first strategy is a continuous variable, and the improvement under the second strategy is a fixed amount. These problems are NP-hard for the general graph. When the graph is a tree, we present an O(|V |) time algorithm to solve the eccentricity lowering problem under the continuous upgrading strategy. However, the problem under the discrete upgrading strategy is still NP-hard even if the graph is a line.