Inverse 1-median Problem Research Articles

Inverse median location problems with variable coordinates

Given n points in $${\mathbb{R}^d}$$ 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 $${\mathbb{R}^d}$$ 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 $${\mathcal{NP}}$$ -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 $${\mathcal{NP}}$$ -hard. Another pseudo-polynomial algorithm is developed for this case, but it is shown that no fully polynomial time approximation scheme does exist.

The inverse 1-median problem on a cycle

Let the graph G = ( V , E ) be a cycle with n + 1 vertices, non-negative vertex weights and positive edge lengths. The inverse 1-median problem on a cycle consists in changing the vertex weights at minimum cost so that a prespecified vertex becomes the 1-median. All cost coefficients for increasing or decreasing the weights are assumed to be 1. We show that this problem can be formulated as a linear program with bounded variables and a special structure of the constraint matrix: the columns of the linear program can be partitioned into two classes in which they are monotonically decreasing. This allows one to solve the problem in O ( n 2 ) time.

Inverse median problems

The inverse p-median problem consists in changing the weights of the customers of a p-median location problem at minimum cost such that a set of p prespecified suppliers becomes the p-median. The cost is proportional to the increase or decrease of the corresponding weight. We show that the discrete version of an inverse p-median problem can be formulated as a linear program. Therefore, it is polynomially solvable for fixed p even in the case of mixed positive and negative customer weights. In the case of trees with nonnegative vertex weights, the inverse 1-median problem is solvable in a greedy-like fashion. In the plane, the inverse 1-median problem can be solved in O(n log n) time, provided the distances are measured in l 1- or l ∞-norm, but this is not any more true in R 3 endowed with the Manhattan metric.