FPTAS’s for Some Cut Problems in Weighted Trees
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.
KeywordsGraph Cut FPTAS Tree Tree Knapsack
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- 8.Xiao, M., Fukunaga, T., Nagamochi, H.: Optimally Cutting a Prescribed Number of Vertices Away From a Tree (manuscript) (2010)Google Scholar