The Inverse $p$-maxian Problem on Trees with Variable Edge Lengths

Abstract

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. Gassner proposed in $2008$ an efficient combinatorial algorithm to solve the inverse $1$-maxian problem on trees. For the case $p \geq 2$, we claim that the problem can be reduced to $O(p^2)$ many inverse $2$-maxian problems. We then develop algorithms to solve the inverse $2$-maxian problem under 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, the problem can be solved in linear time. We also develop $O(n \log n)$ algorithms to solve the problems under Chebyshev norm and bottleneck Hamming distance, where $n$ is the number of vertices of the tree.