Abstract
We consider problems where we are given a rooted tree as input, and must find a subtree with the same root, optimizing some objective function of the nodes in the subtree. When the objective is the sum of linear function weights of a parameter, we show how to list all optima for all parameter values in O(nlogn) time. This can be used to solve many bicriterion optimizations problems in which each node has two values x i and y i associated with it, and the objective function is a bivariate function f(∑x i ,∑y i ) of the sums of these two values. When f is the ratio of the two sums, we have the Weighted Maximum-Mean Subtree Problem, or equivalently the Fractional Prize-Collecting Steiner Tree Problem on Trees; we provide a linear time algorithm when all values are positive, improving a previous O(nlogn) solution, and prove NP-completeness when certain negative values are allowed.
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Carlson, J., Eppstein, D. (2006). The Weighted Maximum-Mean Subtree and Other Bicriterion Subtree Problems. In: Arge, L., Freivalds, R. (eds) Algorithm Theory – SWAT 2006. SWAT 2006. Lecture Notes in Computer Science, vol 4059. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11785293_37
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DOI: https://doi.org/10.1007/11785293_37
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-35753-7
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