Unified approach to the inverse median location problem under the sum and the max objectives applying ordered median function

In this paper, we propose a unified approach to solving the inverse 1-median problem on trees under both the sum and max objectives by using the ordered median function. For the problem under the k-centrum cost function, we investigate the structure of an optimal solution and transform the problem into a univariate optimization problem where the parameter equals the k largest cost. We take advantage of the convexity of the objective function to efficiently find the optimal solution in $ O(n\log n) $ O(nlogn) time, where n is the number of vertices in the tree. For the problem under the centdian cost function, we parameterize it based on the maximum cost item and prove that the induced cost function is convex. We leverage this insight to develop a combinatorial algorithm that solves the corresponding problem in $ O(n\log n) $ O(nlogn) time.