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

TL;DR

This paper develops an instance-dependent algorithm selection framework for the Winner Determination Problem in combinatorial auctions, using graph neural networks and structural features to predict instance hardness and improve solver performance.

Contribution

It introduces a novel hardness prediction method with a lightweight classifier and a GNN specialist for hard instances, enabling effective hybrid solver strategies.

Findings

Hardness classifier achieves 94.7% accuracy in predicting greedy optimality gap.

GNN specialist reduces optimality gap to near zero on hard instances.

Hybrid approach achieves 0.51% overall gap, outperforming pure greedy or GNN methods.

Abstract

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 \emph{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 \emph{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 ${\approx}0\%$ 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 \emph{when} to deploy expensive solvers is more tractable than learning to replace them.