Optimal Algorithms for Integer Inverse Undesirable p-Median Location Problems on Weighted Extended Star Networks

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 $$ {\mathcal {O}}\left( p^2 n \log n\right) $$ O p 2 n log n and $${{\mathcal {O}}}\left( p^{2}(n \log n+ n \log \eta _{\max })\right) $$ O p 2 ( n log n + n log η 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 $$ {\mathcal {O}}(p^2n \log n)$$ O ( p 2 n log n ) time.