We show that the Vehicle Routing Problem (VRP) can be reformulated as a Graph Edit Distance (GED) maximization problem. Under a simple edge-deletion cost model, minimizing total route cost is equivalent to maximizing the total weight of edges deleted from the complete instance graph. This formulation models VRP at the edge level, where solutions are defined by selected edges rather than route sequences, enabling structural analyses that are difficult in classical formulations: per-edge attribution of solution quality, decomposition of the optimality gap, characterization of solution sparsity, and identification of edges that are hard to reach by greedy construction. Theoretically, we establish a merge-decomposition theorem showing that Clarke-Wright savings equal per-merge GED increments, and an approximation-transfer theorem that turns GED approximation ratios into VRP cost bounds. Using this reformulation, we analyze 90 CVRP benchmark instances with known optimal solutions. We find that optimal routing graphs use only 5.5% of available edges, that approximately 3.0% of optimal edges are consistently not found by Clarke-Wright heuristics under repeated restarts, and that the cost gap decomposes into missed optimal edges and substituted non-optimal edges of comparable total weight. The edge-additive objective provides a natural per-edge supervision signal for future graph neural network approaches to edge prediction, suggesting a potential connection to graph neural network approaches that we leave for follow-up work.

翻译：我们证明了车辆路径问题（VRP）可重新表述为图编辑距离（GED）最大化问题。在简单边删除代价模型下，最小化总路径成本等价于从完整实例图中删除边的总权重最大化。该表述在边层级对VRP进行建模——解由选定边而非路径序列定义——从而能够实现经典表述难以进行的结构分析：解质量的逐边归因、最优间隙的分解、解稀疏性的刻画，以及贪婪构造难以触及的边识别。理论上，我们建立了合并-分解定理，证明Clarke-Wright节约值等于逐次合并的GED增量，并提出了近似传递定理，将GED近似比转化为VRP成本边界。利用该重构，我们分析了90个已知最优解的CVRP基准实例，发现最优路由图仅使用5.5%的可用边，约3.0%的最优边在多次重启的Clarke-Wright启发式下始终未被发现，且成本间隙由缺失最优边与替代非最优边（两者总权重相当）共同构成。这种边可加目标函数为未来基于图神经网络的边预测方法提供了天然的逐边监督信号，揭示了与图神经网络方法的潜在关联，我们将此留作后续工作。