Combinatorial algorithms for inverse absolute and vertex 1‐center location problems on trees

Abstract

Abstract. 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 2 ) 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(n 2 r) 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 2 ) time complexity by balancing the height of two trees. In the general case one gets the improved O(n 2 r v ) time complexity where the parameter r v is bounded by n.

The inverse convex ordered 1-median problem on trees under Chebyshev norm and Hamming distance

“…Thus, it is interesting to study in which cases the inverse 1-center problem can be solved in polynomial time. The inverse 1-center problem on unweighted trees was solved efficiently in polynomial time, see [1,2,3]. However, for the inverse 1-center problem on cactus graphs which is a simple generalization of the corresponding problem on tree networks, Nguyen and Chassein [21] showed the NP-hardness.…”

Inverse p-median problems with variable edge lengths

“…They derived an O(n 2 log n) time algorithm for this problem where it is assumed that all the modified edge lengths remain positive. Alizadeh and Burkard [1] investigated the inverse absolute and vertex 1-center location problem on trees with variable edge lengths. They showed that the absolute and vertex 1-center problem can be solved in O(n 2 ) time provided that all edge lengths are strictly positive.…”

A model for the inverse 1-median problem on trees under uncertain costs

“…First, the subtree induced by {v 1 } violates the optimality criterion. Therefore, we have to reduce the weight of v 1 …”

“…Therefore, it is interesting to consider some cases where the inverse 1-center problem can be solved in polynomial time. The inverse 1-center problem on unweighted trees with variable edge lengths was investigated in depth and efficiently solved, see Alizadeh and Burkard [1][2][3]. Nguyen and Chassein [15] …”

“…It can be done in linear time by the algorithm of Balas and Zemel [4]. Now we construct an uncertain variable ϕ corresponding to the minimum cost of (IMUC) with each confidence level α ∈ [0, 1], i.e., {C α } α∈ [0,1] , as follows.…”