Title: Learning to Solve Combinatorial Problems via Efficient Exploration. Abstract: From logistics to the natural sciences, combinatorial optimisation on graphs underpins numerous real-world applications.  Reinforcement learning (RL) has shown particular promise in this setting as it can adapt to specific problem structures and does not require pre-solved instances for these, often NP-hard, problems.  However, state-of-the-art (SOTA) approaches typically suffer from severe scalability issues, primarily due to their reliance on expensive graph neural networks (GNNs) at each decision step.  We introduce ECORD; a novel RL algorithm that alleviates this expense by restricting the GNN to a single pre-processing step, before entering a fast-acting exploratory phase directed by a recurrent unit.  Experimentally, we demonstrate that ECORD achieves a new SOTA for RL algorithms on the Maximum Cut problem, whilst also providing orders of magnitude improvement in speed and scalability.  Compared to the nearest competitor, ECORD reduces the optimality gap by up to 73% on 500 vertex graphs with a decreased wall-clock time.  Moreover, ECORD retains strong performance when generalising to larger graphs with up to 10000 vertices.