The weighted maximum-mean subtree and other bicriterion subtree problems

Authors:
Josiah Carlson;David Eppstein
Affiliations:
Computer Science Department, University of California, Irvine, CA;Computer Science Department, University of California, Irvine, CA
Venue:
SWAT'06 Proceedings of the 10th Scandinavian conference on Algorithm Theory
Year:
2006

Citing 4
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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 xi and yi associated with it, and the objective function is a bivariate function f(∑xi,∑yi) 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