Minimum Equivalent Precedence Relation Systems

TL;DR

This paper investigates the complexity of simplifying systems of linear inequalities representing precedence relations, providing NP-hardness results, polynomial-time solutions under certain conditions, and a decomposition approach for these problems.

Contribution

It introduces a novel analysis of precedence relation system simplification, establishing NP-hardness, polynomial solvability conditions, and a decomposition framework for these problems.

Findings

First problem is NP-hard but solvable under certain conditions.

Second problem is solvable in polynomial time with full solution parameterization.

Generalizes previous results for unweighted directed graphs to linear inequalities.

Abstract

In this paper two related simplification problems for systems of linear inequalities describing precedence relation systems are considered. Given a precedence relation system, the first problem seeks a minimum subset of the precedence relations (i.e., inequalities) which has the same solution set as that of the original system. The second problem is the same as the first one except that the ``subset restriction'' in the first problem is removed. This paper establishes that the first problem is NP-hard. However, a sufficient condition is provided under which the first problem is solvable in polynomial-time. In addition, a decomposition of the first problem into independent tractable and intractable subproblems is derived. The second problem is shown to be solvable in polynomial-time, with a full parameterization of all solutions described. The results in this paper generalize those in [Moyles and Thompson 1969, Aho, Garey, and Ullman 1972] for the minimum equivalent graph problem and transitive reduction problem, which are applicable to unweighted directed graphs.