Inverse median location problems with variable coordinates

Given n points in 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 becomes the 1-median. The cost is proportional to the increase or decrease of the corresponding point coordinate. If the distances between points are measured by the rectilinear norm, the inverse 1-median problem is -hard, but it can be solved in pseudo-polynomial time. Moreover, a fully polynomial time approximation scheme exists in this case. 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 -hard. Another pseudo-polynomial algorithm is developed for this case, but it is shown that no fully polynomial time approximation scheme does exist.