Acyclic orientations with path constraints

TL;DR

This paper introduces a linear programming approach to acyclic orientations with path constraints, connecting them to vertex coloring and frequency assignment problems, and analyzes the associated polytope for optimization.

Contribution

It presents a novel LP formulation for acyclic orientations with path constraints and studies its polytope, including facet-defining constraints and valid inequalities.

Findings

LP formulation effectively models acyclic orientations with path constraints

Polytope analysis reveals facet-defining inequalities

Application to vertex coloring and frequency assignment problems

Abstract

Many well-known combinatorial optimization problems can be stated over the set of acyclic orientations of an undirected graph. For example, acyclic orientations with certain diameter constraints are closely related to the optimal solutions of the vertex coloring and frequency assignment problems. In this paper we introduce a linear programming formulation of acyclic orientations with path constraints, and discuss its use in the solution of the vertex coloring problem and some versions of the frequency assignment problem. A study of the polytope associated with the formulation is presented, including proofs of which constraints of the formulation are facet-defining and the introduction of new classes of valid inequalities.