The Parametric Closure Problem

TL;DR

The paper introduces the parametric closure problem, focusing on computing minimum-weight lower sets in partially ordered sets with linearly varying weights, and provides polynomial solutions for various special cases.

Contribution

It defines the parametric closure problem and offers polynomial time algorithms for important subclasses, generalizing previous results on related problems.

Findings

Polynomial algorithms for semiorders and bounded-treewidth graphs

Efficient solutions for partial orders of bounded width

Generalization of bicriterion subtree problem results

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 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.