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 ≥ 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 l1-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.