The Weighted Maximum-Mean Subtree and Other Bicriterion Subtree Problems

TL;DR

This paper introduces efficient algorithms for bicriterion subtree problems, including a linear-time solution for the weighted maximum-mean subtree with positive weights and complexity results for cases with negative weights.

Contribution

It presents a linear-time algorithm for the weighted maximum-mean subtree problem with positive weights and analyzes NP-completeness when negative weights are involved.

Findings

Linear-time algorithm for positive-weight case

NP-completeness with negative weights

Efficient parametric optimization for bicriterion problems

Abstract

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.