A Quantum Photonic Approach to Graph Coloring

TL;DR

The paper addresses graph coloring by recasting it as an independent-set problem and leveraging Gaussian Boson Sampling (GBS) to identify dense subgraphs and cliques in complement graphs. It introduces GBSC, a quantum-inspired coloring pipeline that uses FindCliques and BestClique to extract large cliques via GBS and seeds a grow-and-swap coloring process, with an exact branch-and-price baseline for comparison. Across Erdős–Rényi graphs and a smart-charging instance, GBSC often achieves competitive or superior colorings, particularly in dense graphs, underscoring the potential of quantum sampling to enhance combinatorial optimization. Acknowledging that results come from classical simulations, the work highlights future scaling opportunities as photonic hardware matures to tackle larger, more complex instances.

Abstract

Gaussian Boson Sampling (GBS) is a quantum computational model that leverages linear optics to solve sampling problems believed to be classically intractable. Recent experimental breakthroughs have demonstrated quantum advantage using GBS, motivating its application to real-world combinatorial optimization problems. In this work, we reformulate the graph coloring problem as an integer programming problem using the independent set formulation. This enables the use of GBS to identify cliques in the complement graph, which correspond to independent sets in the original graph. Our method is benchmarked against classical heuristics and exact algorithms on two sets of instances: Erdős-Rényi random graphs and graphs derived from a smart-charging use case. The results demonstrate that GBS can provide competitive solutions, highlighting its potential as a quantum-enhanced heuristic for graph-based optimization.

Figures (2)

• Figure 1: An illustration of Theorem \ref{['theo:kcoloring_kclique']} with a 7-vertex graph example
• Figure 2: Iterative coloring with GBSC. The first iteration starts from the input graph. This graph is transformed into its $k$-augmented version (here $k=3$ is the ceiling of the Hoffman bound), and then into the complemented augmented graph. The central step, shown in the square, is the identification of cliques via FindCliques, with the most significant one (BestClique) highlighted in red. The resulting $k$-coloring colors all vertices except $\{1,2,8\}$. Since uncolored vertices remain, a second iteration is performed on their induced subgraph, following the same steps (here with $k=2$). The final 5-coloring of the original graph, obtained by combining both iterations, is shown on the right.

Theorems & Definitions (12)

• Definition 1
• Proposition 1
• Definition 2
• Proposition 2
• proof
• Definition 3
• Lemma 1
• proof
• Theorem 1
• proof