Combinatorial algorithms for inverse absolute 1-center location problems on trees

In an inverse network absolute (or vertex) 1-center location problem the parameters of a given network, like edge lengths or vertex weights, have to be modified at minimum total cost such that a prespecified vertex s becomes an absolute (or a vertex) 1-center of the network. In this article, the inverse absolute and vertex 1-center location problems on unweighted trees with n + 1 vertices are considered where the edge lengths can be changed within certain bounds. For solving these problems a fast method is developed for reducing the height of one tree and increasing the height of a second tree under minimum cost until the heights of both trees become equal. Using this result a combinatorial O(n) time algorithm is stated for the inverse absolute 1-center location problem in which no topology change occurs. If topology changes are allowed, an O(nr) time algorithm solves the problem where r, r < n, is the compressed depth of the tree network T rooted in s. Finally, the inverse vertex 1-center problem with edge length modifications is solved on T . If all edge lengths remain positive this problem can be solved within the improved O(n) time complexity by balancing the height of two trees. In the general case one gets the improved O(nrv) time complexity where the parameter rv is bounded by n.