Finding a length-constrained maximum-sum or maximum-density subtree and its application to logistics

We study the problem of finding a length-constrained maximum-density path in a tree with weight and length on each edge. This problem was proposed in [R.R. Lin, W.H. Kuo, K.M. Chao, Finding a length-constrained maximum-density path in a tree, Journal of Combinatorial Optimization 9 (2005) 147–156] and solved in O(nU)O(nU) time when the edge lengths are positive integers, where nn is the number of nodes in the tree and UU is the length upper bound of the path. We present an algorithm that runs in O(nlog2n)O(nlog2n) time for the generalized case when the edge lengths are positive real numbers, which indicates an improvement when U=Ω(log2n)U=Ω(log2n). The complexity is reduced to O(nlogn)O(nlogn) when edge lengths are uniform. In addition, we study the generalized problems of finding a length-constrained maximum-sum or maximum-density subtree in a given tree or graph, providing algorithmic and complexity results.