A depth‐first dynamic programming procedure for the extended tree knapsack problem in local access network design

The Extended Tree Knapsack Problem (ETKP) is a generalized version of the Tree Knapsack Problem where an arbitrary nonlinear traffic‐flow cost is imposed. This problem can be solved by the straight‐forward “bottom‐up” approach with a time complexity of O(nH2), where n is the number of nodes in the tree, and H is the knapsack capacity. In this paper, we show that if the traffic‐flow cost function is the cable expansion cost, which occurs in the Local Access Telecommunication Network (LATN) expansion, this special ETKP can be solved by a depth‐first dynamic programming procedure in a time complexity of O(nδH), where δ is the largest existing cable capacity in LATN. This result indicates that the depth‐first dynamic programming algorithm can be applied for solving a general class of tree optimization problems. The computational results of our algorithm for the ETKP are also provided.

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