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.
Upgrading p-median problem is a problem of finding the best median of the given graph through modification of its parameters. The current paper develops a polynomial-time model to address this problem when the weights of vertices can be varying under a given budget. Moreover, in the case where the considering graph has a special structure, namely a path, a linear time algorithm will be proposed for solving the problem with uniform cost.
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