SAG-Sep: Sparse Augmented Graphs and Onion-Guided Search for Rounded Capacity Cut Separation

Haoran Liu, Guanyi Wang, Yu Yang

Rounded capacity cuts (RCCs) are among the most effective cuts for capacitated vehicle-routing relaxations, but exact separation is computationally prohibitive at scale. Existing heuristic and neural separators can be fast, but often fail to produce sufficiently many effective cuts within practical separation budgets. We introduce SAG-Sep, a separator that scores node membership in RCC-inducing subsets across vehicle-count levels from a fractional LP solution in a single encoder forward pass, avoiding the iterative predict-and-coarsen inference used by NeuralSEP-style separators. SAG-Sep encodes the LP solution as a Sparse Augmented Graph (SAG), adding flow-informed multi-hop edges that expose long-range routing structure without densifying attention. Beyond LP-aware sparse encoding, we observe that high-value exact RCCs are typically nested, organizing into onion-like subset families, and SAG-Sep's node scores track subset nesting depth. This motivates Onion-Sweep (OS), a training-free decoder that ranks customers by their scores, evaluates the RCC violation of every prefix of the ranking in one vectorized pass, and emits the most violated prefix of each replica together with a…