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The inverse $p$-maxian problem on trees with variable edge lengths

We concern the problem of modifying the edge lengths of a tree in minimum total cost so that the prespecified $p$ vertices become the $p$-maxian with respect to the new edge lengths. This problem is called the inverse $p$-maxian problem on trees. \textbf{Gassner} proposed efficient combinatorial alogrithm to solve the the inverse 1-maxian problem on trees in 2008. For the problem with $p \geq 2$, we claim that the problem can be reduced to finitely many inverse $2$-maxian problem. We then develop algorithms to solve the inverse $2$-maxian problem for various objective functions. The problem under $l_1$-norm can be formulated as a linear program and thus can be solved in polynomial time. Particularly, if the underlying tree is a star, then the problem can be solved in linear time. We also devised $O(n\log n)$ algorithms to solve the problems under Chebyshev norm and bottleneck Hamming distance, where $n$ is the number of vertices of the tree. Finally, the problem under weighted sum Hamming distance is $NP$-hard.