Center problems with pos/neg weights on trees

In a network with positive and negative vertex weights the pos/neg 1-center problem asks to minimize a linear combination of the maximum weighted distances of the center to the vertices with positive weights and to the vertices with negative weights, respectively. We show that in a network with n vertices and m edges the pos/neg 1-center problem can be solved in O(mnlogn) time. In trees a better complexity can be achieved. In the case of a path or of a star graph this problem can be solved in linear time. Further this problem is studied for a cactus with vertex weights 1 and −1. Moreover, an algorithm for the discrete anti-p-center problem on a tree with the improved time complexity O(nlog²n) is given. Finally, the pos/neg discrete p-center on a tree is treated and solved by an algorithm of time complexity O(n²logn).