Title: LOMAC: GNN-based Deep Reinforcement Learning with One-Way Markov Chain for Graph Coloring
Primary area: optimization
Keywords: Deep Reinforcement Learning, Graph Coloring Problem, Graph Neural Network, One-way Markov Chain

Abstract: The graph coloring problem (GCP) is an NP-hard combinatorial optimization task aimed at assigning the minimum number of colors to graph vertices such that no two adjacent vertices share the same color. While deep reinforcement learning (DRL) and graph neural networks (GNNs) are promising approaches to solving the GCP, their scalability is usually limited by the large number of Markov states and high computational complexity as the graph size increases. In this paper, we introduce LOMAC, a novel GNN-based DRL framework that integrates a one-way, two-dimensional Markov chain and a linear-complexity GNN model with pseudonode-enhanced message passing. This integration significantly reduces both space and computational complexity. We transform the GCP into a one-way Markov chain model, introducing two key concepts: Markov state potential and graph state potential. Through theoretical analysis of Markov- and graph-state potentials, we effectively guide the search for an optimal vertex-coloring solution. We show that LOMAC reduces the number of Markov states from $O(K^N)$ to $O(NK)$, simplifying decision-making with unidirectional state transitions. Additionally, an invalid action penalty mechanism is implemented to further optimize the coloring process. Experimental results in various sizes of \textit{Erdős–Rényi}- and \textit{Barabási–Albert} graphs and 16 real-world benchmarks demonstrate that LOMAC achieves state-of-the-art performance in the number of required colors.

TL;DR: A novel GNN-based DRL framework that integrates a one-way, two-dimensional Markov chain and a linear-complexity dynamic message-passing GNN model for efficient graph coloring.