Title: The Vehicle Routing Problem as a Graph Edit Distance Maximization Task: Theoretical Insights and Empirical Analysis

This study demonstrates that the Vehicle Routing Problem (VRP) can be effectively reframed as a Graph Edit Distance (GED) maximization challenge. By employing a straightforward edge-deletion cost framework, we establish that minimizing the overall cost of vehicle routes is mathematically equivalent to maximizing the cumulative weight of edges removed from the complete instance graph. This edge-centric formulation defines solutions through the selection of specific edges rather than the ordering of routes, thereby facilitating structural analyses that are often intractable in traditional models. These analyses include attributing solution quality on a per-edge basis, decomposing the optimality gap, characterizing the sparsity of solutions, and pinpointing edges that prove difficult for greedy construction methods to identify.

On a theoretical level, we derive a merge-decomposition theorem which proves that Clarke-Wright savings correspond directly to per-merge GED increments. Additionally, we present an approximation-transfer theorem, which allows GED approximation ratios to be converted into bounds for VRP costs. Leveraging this reformulation, we conducted an analysis of 90 CVRP benchmark instances with established optimal solutions. Our findings reveal that optimal routing configurations utilize merely 5.5% of the available edges. Furthermore, we observed that roughly 3.0% of optimal edges are consistently overlooked by Clarke-Wright heuristics, even after multiple restarts. The analysis also shows that the cost gap splits into two components of comparable total weight: the omission of optimal edges and the inclusion of non-optimal substituted edges. Finally, the edge-additive nature of the objective function offers a natural per-edge supervision signal for upcoming graph neural network methods focused on edge prediction, hinting at a potential link to GNN approaches that we reserve for subsequent investigation.

Source: arXiv Generated at: 2026-06-02 00:00:00 UTC