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Fractional Chromatic Numbers from Exact Decision Diagrams

Timo Brand, Stephan Held

TL;DR

This work proves that the solution to the linear flow relaxation on exact decision diagrams determines the fractional chromatic number of a graph, and establishes that the integrality gap of the linear programming relaxation is O(log n), where n represents the number of vertices in the graph.

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.

Fractional Chromatic Numbers from Exact Decision Diagrams

TL;DR

This work proves that the solution to the linear flow relaxation on exact decision diagrams determines the fractional chromatic number of a graph, and establishes that the integrality gap of the linear programming relaxation is O(log n), where n represents the number of vertices in the graph.

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.

Paper Structure

This paper contains 8 sections, 3 theorems, 4 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Contributions
  3. Decision Diagrams
  4. Decision Diagrams for Stable Sets
  5. Graph Coloring from Stable Set Decision Diagrams
  6. The Fractional Chromatic Number and Decision Diagrams
  7. Computational Results with Exact Arithmetic
  8. Conclusions

Key Result

theorem 1

Let $G=(V,E)$ be a graph and $D=(N,A)$ an exact stable set decision diagram for $G$. Then the linear relaxation $(F')$ is equivalent to (VCLP).

Figures (1)

  • Figure 1: An example graph and its exact decision diagram.

Theorems & Definitions (5)

  • theorem 1
  • proof
  • corollary 1
  • proof
  • corollary 2