加权最大均值子树及其它双临界子树问题
The Weighted Maximum-Mean Subtree and Other Bicriterion Subtree Problems
摘要：
我们考虑这样的问题，即给定一棵有根树作为输入，并且必须找到一个具有相同根的子树，优化子树中节点的目标函数。当这个函数是常数节点权重之和时，问题在线性时间内得到了平凡的解决。当目标函数是参数的线性函数的权值之和时，我们展示了如何在O（nlog n）时间内列出所有可能的参数值的所有最优解;该参数优化问题可用于解决许多双临界优化问题，其中每个节点具有与其相关联的两个值xi和yi，并且目标函数是双变量函数f（SUM（xi），SUM（yi））这两个值之和。一个特殊的情况下，当f是两个总和的比率，是加权最大均值子树问题，或等价的分数奖金收集Steiner树问题的树;对于这种特殊情况下，我们提供了一个线性时间算法，这个问题的所有权重是积极的，改善以前的O（n log n）的解决方案，并证明了该问题是NP-完全时，允许负权重。
英文摘要：
We consider problems in which 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 this function is the sum of constant node weights, the problem is trivially solved in linear time. When the objective is the sum of weights that are linear functions of a parameter, we show how to list all optima for all possible parameter values in O(n log n) time; this parametric optimization problem can be used to solve many bicriterion optimizations problems, in which each node has two values xi and yi associated with it, and the objective function is a bivariate function f(SUM(xi),SUM(yi)) of the sums of these two values. A special case, when f is the ratio of the two sums, is the Weighted Maximum-Mean Subtree Problem, or equivalently the Fractional Prize-Collecting Steiner Tree Problem on Trees; for this special case, we provide a linear time algorithm for this problem when all weights are positive, improving a previous O(n log n) solution, and prove that the problem is NP-complete when negative weights are allowed.
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