Inverse 1-center location problems with edge length augmentation on trees

Abstract

This paper considers the inverse 1-center location problem with edge length augmentation on a tree network T with n + 1 vertices. The goal is to increase the edge lengths at minimum total cost subject to given modification bounds such that a prespecified vertex s becomes an absolute 1-center under the new edge lengths. Using a set of suitably extended AVL-search trees we develop a combinatorial algorithm which solves the inverse 1-center location problem with edge length augmentation in O(n log n) time. Moreover, it is shown that the problem can be solved in O(n) time if all the cost coefficients are equal.

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.…”

The inverse 1-center problem on trees with variable edge lengths under Chebyshev norm and Hamming distance

“…They also developed an O(cn log n) time algorithm for uniform-cost inverse 1-center problem on trees where c is the compressed depth of the tree. Furthermore, Alizadeh et al (2009) provided an O(n log n) algorithm to solve the inverse 1-center problem on (unweighted) trees with edge length augmentation. The complexity was improved to linear time in case the costs of edge lengths modification are uniform.…”

Inverse p-median problems with variable edge lengths

“…Dropping this condition, they proposed an O(n 2 r) time algorithm where the parameter r bounded by ⌈ n 2 ⌉. Recently Alizadeh, Burkard and Pferschy [2] used a set of suitably extended AVL-search trees and developed a combinatorial algorithm which solves the inverse 1-center location problem with edge length augmentation in O(n log n) time. In this article we investigate the inverse p-median problem with variable edge lengths.…”