Precedence-Constrained Arborescences

TL;DR

This paper proves the NP-hardness of the precedence-constrained minimum-cost arborescence problem and introduces a scalable MILP model that outperforms previous models, also proposing a new variation of the problem.

Contribution

It establishes NP-hardness for the precedence-constrained arborescence problem and presents a new, more efficient MILP model, along with a novel variation of the problem.

Findings

The new MILP model performs substantially better than previous models.

The precedence-constrained arborescence problem is NP-hard.

A new variation of the problem is introduced and shown to be NP-hard.

Abstract

The minimum-cost arborescence problem is a well-studied problem in the area of graph theory, with known polynomial-time algorithms for solving it. Previous literature introduced new variations on the original problem with different objective function and/or constraints. Recently, the Precedence-Constrained Minimum-Cost Arborescence problem was proposed, in which precedence constraints are enforced on pairs of vertices. These constraints prevent the formation of directed paths that violate precedence relationships along the tree. We show that this problem is NP-hard, and we introduce a new scalable mixed integer linear programming model for it. With respect to the previous models, the newly proposed model performs substantially better. This work also introduces a new variation on the minimum-cost arborescence problem with precedence constraints. We show that this new variation is also NP-hard, and we propose several mixed integer linear programming models for formulating the problem.