Learning Structural Hardness for Combinatorial Auctions: Instance-Dependent Algorithm Selection via Graph Neural Networks

The Winner Determination Problem (WDP) in combinatorial auctions is NP-hard, and no existing method reliably predicts which instances will defeat fast greedy heuristics. The ML-for-combinatorial-optimization community has focused on learning to replace solvers, yet recent evidence shows that graph neural networks (GNNs) rarely outperform well-tuned classical methods on standard benchmarks. We pursue a different objective: learning to predict when a given instance is hard for greedy allocation, enabling instance-dependent algorithm selection. We design a 20-dimensional structural feature vector and train a lightweight MLP hardness classifier that predicts the greedy optimality gap with mean absolute error 0.033, Pearson correlation 0.937, and binary classification accuracy 94.7% across three random seeds. For instances identified as hard -- those exhibiting ``whale-fish'' trap structure where greedy provably fails -- we deploy a heterogeneous GNN specialist that achieves ≈0% optimality gap on all six adversarial configurations tested (vs. 3.75--59.24% for greedy). A hybrid allocator combining the hardness classifier with GNN and greedy solvers achieves 0.51% overall gap on mixed distributions. Our honest evaluation on CATS benchmarks confirms that GNNs do not outperform Gurobi (0.45--0.71 vs. 0.20 gap), motivating the algorithm selection framing. Learning when to deploy expensive solvers is more tractable than learning to replace them.