The inverse 1-maxian problem with edge length modification

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We consider the problem of modifying the lengths of the edges of a graph at minimum cost such that a prespecified vertex becomes a 1-maxian with respect to the new edge lengths. The inverse 1-maxian problem with edge length modification is shown to be strongly \(\mathcal{N}\mathcal{P}\) -hard and remains weakly \(\mathcal{NP}\) -hard even on series-parallel graphs. Moreover, a transformation of the inverse 1-maxian problem with edge length modification on a tree to a minimum cost circulation problem is given which solves the original problem in \(\mathcal{O}(n\log n)\) .