Finding a weight-constrained maximum-density subtree in a tree

Abstract

Given a tree T = (V, E) of n vertices such that each node v is associated with a value-weight pair (valv, wv), where value val v is a real number and weight wv is a non-negative integer, the density of T is defined as ∑v ∈ V val v/∑v ∈ V wv. A subtree of T is a connected subgraph (V′, E′) of T, where V′ ⊆ V and E′ ⊆ E. Given two integers wmin and wmax, the weight-constrained maximum-density subtree problem on T is to find a maximum-density subtree T′ = (V′, E′) satisfying w min ≤ ∑v∈V′ wv ≤ w max. In this paper, we first present an O(wmaxn)-time algorithm to find a weight-constrained maximum-density, path in a tree, and then present an O(wmax2n)-time algorithm to find a weight-constrained maximum-density subtree in a tree. Finally, given a node subset S ⊂ V, we also present an O(wmax2n)-time algorithm to find a weight-constrained maximum-density subtree of T which covers all the nodes in S. © Springer-Verlag Berlin Heidelberg 2005.