The weight-constrained maximum-density subtree problem and related problems in trees

Given a tree T=(V,E) of n nodes such that each node v is associated with a value-weight pair (valv,wv), where valuevalv is a real number and weightwv is a non-negative integer, the density of T is defined as $\frac{\sum_{v\in V}{\mathit{val}}_{v}}{\sum_{v\in V}w_{v}}$. 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 wmin ≤∑v∈V′wv≤wmax . In this paper, we first present an O(wmax n)-time algorithm to find a weight-constrained maximum-density path in a tree T, and then present an O(wmax 2n)-time algorithm to find a weight-constrained maximum-density subtree in T. Finally, given a node subset S⊂V, we also present an O(wmax 2n)-time algorithm to find a weight-constrained maximum-density subtree in T which covers all the nodes in S.