The Parametric Closure Problem

Abstract

Abstract. 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 downsets 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.

The Graphs of Stably Matchable Pairs

“…Several of our characterizations will relate the lattices of stable matchings to the lattices of closures of a directed graph. In this context, a closure is a subset of the vertices of a given directed graph that has no outgoing edges: there is no edge from a vertex in the closure to a vertex outside the closure [9,24]. The closures of a graph, ordered by subsets, form a distributive lattice, which is the lattice of lower sets in a partial order on strongly connected components of the given graph, defined by setting X ≤ Y whenever a vertex of component X can be reached from a vertex of Y .…”

A Stronger Lower Bound on Parametric Minimum Spanning Trees

“…It can be solved by constructing the sequence of parametric minimum spanning trees and evaluating the combination of weights for each one [22,24]. Beyond spanning trees, other combinatorial optimization problems that have been considered from the same parametric and bicriterion point of view include shortest paths [3][4][5]11], optimal subtrees of rooted trees [2], minimum-weight bases of matroids [9,20], maximum flows and minimum cuts [15,17], minimum-weight closures of directed graphs [10], sequence alignment [18], and the knapsack problem [8,16,21].…”

A Stronger Lower Bound on Parametric Minimum Spanning Trees

“…Any bicriterion optimization problem that can be expressed as maximizing a quasiconvex function (or minimizing a quasiconcave function) of the two kinds of total weight automatically has its optimum at a convex hull vertex, and can be solved by constructing the sequence of parametric minimum spanning trees and evaluating the combination of weights for each one [18]. Other combinatorial optimization problems that have been considered from the same parametric and bicriterion point of view include shortest paths [3][4][5]11], optimal subtrees of rooted trees [2], minimum-weight bases of matroids [9], minimum-weight closures of directed graphs [10], and the knapsack problem [8,15,17].…”