Abstract
Given a tree with nonnegative edge cost and nonnegative vertex weight, and a number k ≥ 0, we consider the following four cut problems: cutting vertices of weight at most or at least k from the tree by deleting some edges such that the remaining part of the graph is still a tree and the total cost of the edges being deleted is minimized or maximized. The MinMstCut problem (cut vertices of weight at most k and minimize the total cost of the edges being deleted) can be solved in linear time and space and the other three problems are NP-hard. In this paper, we design an O(ln/ε)-time O(l 2/ε + n)-space algorithm for MaxMstCut, and O(ln(1/ε + logn))-time O(l 2/ε + n)-space algorithms for MinLstCut and MaxLstCut, where n is the number of vertices in the tree, l the number of leaves, and ε> 0 the prescribed error bound.
This work was partially supported by Grant-in-Aid for Scientific Research from the Ministry of Education, Culture, Sports, Science and Technology of Japan, and National Natural Science Foundation of China under the Grant 60903007. Part of the work was done when the first author was visiting Nagamochi’s lab in Kyoto University.
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Xiao, M., Fukunaga, T., Nagamochi, H. (2010). FPTAS’s for Some Cut Problems in Weighted Trees. In: Lee, DT., Chen, D.Z., Ying, S. (eds) Frontiers in Algorithmics. FAW 2010. Lecture Notes in Computer Science, vol 6213. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-14553-7_21
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DOI: https://doi.org/10.1007/978-3-642-14553-7_21
Publisher Name: Springer, Berlin, Heidelberg
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