The inverse 1-median problem on a cycle

Let the graph G = (V,E) be a cycle with n + 1 vertices, nonnegative vertex weights and positive edge lengths. The inverse 1-median problem on a cycle consists in changing the vertex weights at minimum cost such that a prespecified vertex becomes the 1-median. The cost is proportional to the increase or decrease of the corresponding weight. 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 to solve the problem in O(n)-time.