Abstract: This paper is concerned with the problem of modifying the edge lengths of a weighted extended star network with n vertices by integer amounts at the minimum total cost subject to be given modification bounds so that a set of p prespecified vertices becomes an undesirable p-median location on the perturbed network. We call this problem as the integer inverse undesirable p-median location model. Exact combinatorial algorithms with $\mathcal{O}({p^2}\;n\;\log \;n)$ and $\mathcal{O}({p^2}(n\;\log \;n + n\;\log \;\eta {}_{\max }))$ running times are proposed for solving the problem under the weighted rectilinear and weighted Chebyshev norms, respectively. Furthermore, it is shown that the problem under the weighted sum-type Hamming distance with uniform modification bounds can be solved in $\mathcal{O}({p^2}\;n\;\log \;n)$ time.

Key words: Undesirable p-median location, Combinatorial optimization, Inverse optimization, Time complexity