Fractional Chromatic Numbers from Exact Decision Diagrams

TL;DR

This paper proves that the linear flow relaxation on exact decision diagrams computes the fractional chromatic number of a graph, providing new insights into graph coloring bounds and solving previously unknown instances.

Contribution

It establishes that the flow relaxation solution determines the fractional chromatic number and shows the integrality gap is O(log n), with practical experiments solving complex benchmark instances.

Findings

Flow relaxation determines fractional chromatic number

Integrality gap is O(log n)

Solved previously unknown DIMACS instance

Abstract

Recently, Van Hoeve proposed an algorithm for graph coloring based on an integer flow formulation on decision diagrams for stable sets. We prove that the solution to the linear flow relaxation on exact decision diagrams determines the fractional chromatic number of a graph. This settles the question whether the decision diagram formulation or the fractional chromatic number establishes a stronger lower bound. It also establishes that the integrality gap of the linear programming relaxation is O(log n), where n represents the number of vertices in the graph. We also conduct experiments using exact decision diagrams and could determine the chromatic number of r1000.1c from the DIMACS benchmark set. It was previously unknown and is one of the few newly solved DIMACS instances in the last 10 years.