An efficient algorithm for finding the semi-obnoxious (k,l) -core of a tree

In this paper we study finding the (k,l)-core problem on a tree which the vertices have positive or negative weights. Let T=(V,E) be a tree. The (k,l)-core of T is a subtree with at most k leaves and with a diameter of at most l which the sum of the weighted distances from all vertices to this subtree is minimized. We show that, when the sum of the weights of vertices is negative, the (k,l)-core must be a single vertex. Then we propose an algorithm with time complexity of O(n2logn) for finding the (k,l) -core of a tree with pos/neg weight, which is in fact a modification of the one proposed by Becker et al. [Networks 40 (2002) 208].