4 research outputs found
The Weighted Maximum-Mean Subtree and Other Bicriterion Subtree Problems
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.Comment: 10 page
The Parametric Closure Problem
We define the parametric closure problem, in which the input is a partially
ordered set whose elements have linearly varying weights and the goal is to
compute the sequence of minimum-weight lower sets of the partial order as the
weights vary. We give polynomial time solutions to many important special cases
of this problem including semiorders, reachability orders of bounded-treewidth
graphs, partial orders of bounded width, and series-parallel partial orders.
Our result for series-parallel orders provides a significant generalization of
a previous result of Carlson and Eppstein on bicriterion subtree problems.Comment: 22 pages, 8 figures. A preliminary version of this paper appeared at
the 14th Algorithms and Data Structures Symposium (WADS), Victoria, BC,
August 2015, Springer, Lecture Notes in Comp. Sci. 9214 (2015), pp. 327-33
The weighted maximum-mean subtree and other bicriterion subtree 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 ( β xi, β 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.
The weighted maximum-mean subtree and other bicriterion subtree 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 ( β xi, β 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.