The inverse 1-median problem on a tree and on a path

This article considers the inverse 1-median problem on a tree and on a path. The aim is to change the vertex weights at minimum total cost with respect to given modification bounds such that a prespecified vertex becomes 1-median. The inverse 1-median problem on trees with nonnegative weights can be formulated as a continuous knapsack problem and therefore the problem is solvable in O ( n ) -time. For a path with pos/neg weights the 1-median lies on one of the vertices with positive weights or lies on one of the end points. This property leads us to solve the inverse 1-median problem on a path with negative weights (the weight of endpoints are arbitrary) in linear time.