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

Abstract

This paper addresses the problem of modifying the edge lengths of a tree in minimum total cost such that a prespecified vertex becomes the 1-center of the perturbed tree. This problem is called the inverse 1-center problem on trees. We focus on the problem under Chebyshev norm and Hamming distance. From special properties of the objective functions, we can develop combinatorial algorithms to solve the problem. Precisely, if there does not exist any vertex coinciding with the prespecified vertex during the modification of edge lengths, the problem under Chebyshev norm or bottleneck Hamming distance is solvable in O(n log n) time, where n + 1 is the number of vertices of the tree. Dropping this condition, the problem can be solved in O(n 2 ) time.

The inverse minisum circle location problem

“…Later, Alizadeh and Burkard [3] showed that the inverse 1-center problem can be solved in O(n 2 ) time. The inverse 1-median and 1-center problems on trees with Chebyshev and Hamming norms have been considered by Guan and Zhang [15] and Nguyen and Sepasian [20], respectively. Later, Sepasian and Rahbarnia [23] solved the inverse 1-median problem with varying vertex and edge length on trees.…”

Inverse single facility location problem on a tree with balancing on the distance of server to clients

“…When the underlying network is a cycle, Burkard et al [9] proposed an O(n 2 ) algorithm for the inverse 1-median problem. The inverse 1-median and 1center problems on trees with Chebyshev and Hamming norms have been considered by Guan and Zhang [16] and Nguyen and Sepasian [26], respectively. Sepasian and Rahbarnia [30] investigated the inverse 1-median problem with varying vertex and edge length on trees.…”

Inverse group 1-median problem on trees

“…The inverse center location problem is N P -hard; see Cai et al [8]. However, the corresponding problem on trees is solvable in polynomial time; see the problem on unweighted trees [2,3], the problem on weighted trees [30], the problem under Chebyshev norm and Hamming distance [27]. Moreover, inverse obnoxious location problems were also under study with efficient solution methods; see Alizadeh et al [4,1].…”