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

• Main Authors: LAU, Hoong Chuin, NGO, Trung Hieu, Nguyen, Bao Nguyen
• Format: text
• Language: English
• Published: Institutional Knowledge at Singapore Management University 2006
• Subjects: Network design; Algorithm; Computational complexity; Logistics; Numerical Analysis and Scientific Computing; Operations Research, Systems Engineering and Industrial Engineering
• Online Access: https://ink.library.smu.edu.sg/sis_research/1188; https://ink.library.smu.edu.sg/context/sis_research/article/2187/viewcontent/Finding_a_length_constrained_maximum_sum_2006_afv.pdf
• Institution: Singapore Management University

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) time when the edge lengths are positive integers, where n is the number of nodes in the tree and U is the length upper bound of the path. We present an algorithm that runs in O(nlog2n) time for the generalized case when the edge lengths are positive real numbers, which indicates an improvement when U=Ω(log2n). The complexity is reduced to 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.