Inverse 1-median problem on trees under mixed rectilinear and Chebyshev norms

Abstract

We consider in this paper the inverse 1-median problem on trees with variable vertex weights, which are partitioned into groups. While the modifications in each group are measured by Chebyshev norm, rectilinear norm is applied for computing the cost of groups; and vice versa. As a result, it yields the sum of max and the max of sum objective functions. For the sum of max objective, we develop an O(Mlog⁡M) time algorithm based on a parameterization approach, where M is the number of vertices in the tree. On the other hand, the problem under the max of sum objective can be solved in linear time by an algorithm that prunes a half of indices in each iteration.