Algorithms for path medi-centers of a tree

We consider the problem of finding an optimal location of a path on a tree, using combinations of minisum and minimax criteria (which are respectively maximal distance and average distance from the path to customers situated at the vertices). The case of linear combination of the two criteria and the case where one criterion is optimized subject to a restriction on the value of the other are considered and linear-time algorithms for these problems are presented. It is proved that the representation of the set of Pareto-optimal paths in the space of criteria has cardinality not greater than n – 1, where n is the number of vertices of the tree, and can be obtained in O(n log n) time, although the number of Pareto-optimal paths can be O(n²).