Finding a maximum-density path in a tree under the weight and length constraints

Let T = (V ,E) be a tree with n nodes such that each node v is associated with a value-weight pair (valv,wv), where the value valv is a real number and the weight wv is a positive integer. The density of a path P = 〈v1, v2, . . . , vk〉 is defined as ∑k i=1 vali/ ∑k i=1 wi . The weight of P , denoted by w(P ), is ∑k i=1 wi . Given a tree of n nodes, two integers wmin and wmax, and a length lower bound L, we present a pseudo-polynomial O(wmaxnL)-time algorithm to find a maximum-density path P in T such that wmin w(P ) wmax and the length of P is at least L. © 2007 Elsevier B.V. All rights reserved.