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(n²) time.