Combinatorial algorithms for inverse absolute and vertex 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 problem on trees with n + 1 vertices is 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 height of both trees is 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, a new approach yields the improved O(n) time complexity. In the general case one gets the improved O(nrv) time complexity where the parameter rv is bounded by dn/2e.