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 (val _v,w _v ), where value val _v is a real number and weight w _v is a non-negative integer, the density of T is defined as Σv∈V Valu/Σv∈Vwv. A subtree of T is a connected subgraph (V′,E′) of T, where V′V and E′E. Given two integers w _min∈ and w _max, the weight-constrained maximum-density subtree problem on T is to find a maximum-density subtree T′=(V′,E′) satisfying w _minΣ_{v V′} w _v ≤w _max. In this paper, we first present an O(w _max n)-time algorithm to find a weight-constrained maximum-density path in a tree T, and then present an O(w _max ² n)-time algorithm to find a weight-constrained maximum-density subtree in T. Finally, given a node subset S⊂V, we also present an O(w _max ² n)-time algorithm to find a weight-constrained maximum-density subtree in T which covers all the nodes in S.