UDC: A Unified Neural Divide-and-Conquer Framework for Large-Scale Combinatorial Optimization Problems

TL;DR

The paper tackles the challenge of solving large-scale combinatorial optimization with neural methods by introducing UDC, a unified divide-and-conquer framework with Divide-Conquer-Reunion (DCR) training. UDC uses a fast heatmap-based graph neural division to generate fixed-length subproblems and applies problem-specific conquering solvers, augmented by a Reunion step to repair connections and improve final solutions. The authors demonstrate that unified, joint training of division and conquering policies yields superior performance across 10 diverse CO problems and scales up to very large instances, with significant efficiency gains over baselines. The approach, along with open-source code, suggests broad applicability to general offline CO tasks without heavy reliance on problem-specific heuristics. Overall, UDC advances scalable, generalizable neural CO by integrating division, conquering, and reunion into a single training loop.

Abstract

Single-stage neural combinatorial optimization solvers have achieved near-optimal results on various small-scale combinatorial optimization (CO) problems without requiring expert knowledge. However, these solvers exhibit significant performance degradation when applied to large-scale CO problems. Recently, two-stage neural methods motivated by divide-and-conquer strategies have shown efficiency in addressing large-scale CO problems. Nevertheless, the performance of these methods highly relies on problem-specific heuristics in either the dividing or the conquering procedure, which limits their applicability to general CO problems. Moreover, these methods employ separate training schemes and ignore the interdependencies between the dividing and conquering strategies, often leading to sub-optimal solutions. To tackle these drawbacks, this article develops a unified neural divide-and-conquer framework (i.e., UDC) for solving general large-scale CO problems. UDC offers a Divide-Conquer-Reunion (DCR) training method to eliminate the negative impact of a sub-optimal dividing policy. Employing a high-efficiency Graph Neural Network (GNN) for global instance dividing and a fixed-length sub-path solver for conquering divided sub-problems, the proposed UDC framework demonstrates extensive applicability, achieving superior performance in 10 representative large-scale CO problems. The code is available at https://github.com/CIAM-Group/NCO_code/tree/main/single_objective/UDC-Large-scale-CO-master.

Figures (9)

• Figure 1: The solving process of the proposed UDC on large-scale VRP, 0-1 Knapsack Problem (KP), and Maximum Independent Set (MIS).
• Figure 2: The proposed DCR training method. The upper part shows the pipeline of a single DCR-enabled training step, including a Divide step, a Conquer step, and a Reunion step. The lower part provides a local view of a solution fragment to demonstrate our motivation, the additional Reunion step can repair the wrong connections from sub-optimal sub-problem dividing results. DCR has the potential to correct sub-optimal dividing results by repairing the connection part in the additional Reunion step.
• Figure 3: Training curves of UDC variants with different training schemes.
• Figure 4: The process of preparing sub-CVRPs.
• Figure 5: Testing curves for ablation study.