Inverse median location problems with variable coordinates

Given n + 1 points in Rd with nonnegative weights, the inverse 1-median problem with variable coordinates consists in changing the coordinates of the given points at minimum cost such that a prespecified point in Rd becomes the 1-median. The cost is proportional to the increase or decrease of the corresponding point coordinate. In case that the distances between points are measured by the rectilinear norm, it is shown that the inverse 1-median problem is NP-hard, but it can be solved by a pseudo-polynomial algorithm. If the point weights are assumed to be equal, the corresponding inverse problem can be reduced to d continuous knapsack problems and is therefore solvable in O(nd) time. In case that the squared Euclidean norm is used, we derive another efficient combinatorial algorithm which solves the problem in O(nd) time. It is also shown that the inverse 1-median problem endowed with the Chebyshev norm in the plane is NP-hard. Another pseudo-polynomial algorithm is developed for this case.