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

Sungwoo Kang

TL;DR

This work tackles the Winner Determination Problem (WDP) in combinatorial auctions by reframing ML from solver replacement to instance-aware algorithm selection. A 20-feature hardness classifier predicts greedy failure with $MAE=0.033$, $r=0.937$, and $94.7\%$ accuracy, enabling a practical triage that routes hard instances to a heterogeneous GNN specialist, which achieves near-zero optimality gap on adversarial whale-fish traps. A hybrid allocator combining the hardness predictor, GNN, and greedy solvers attains an overall gap of $0.51\%$ on mixed distributions, while plain GNNs on standard benchmarks like CATS do not beat Gurobi, underscoring the value of instance-dependent solver selection. The findings advocate for a hybrid, structure-aware approach where ML identifies hardness and guides the deployment of traditional optimization techniques, achieving high-quality results with limited computational overhead.

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.

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

TL;DR

This work tackles the Winner Determination Problem (WDP) in combinatorial auctions by reframing ML from solver replacement to instance-aware algorithm selection. A 20-feature hardness classifier predicts greedy failure with , , and accuracy, enabling a practical triage that routes hard instances to a heterogeneous GNN specialist, which achieves near-zero optimality gap on adversarial whale-fish traps. A hybrid allocator combining the hardness predictor, GNN, and greedy solvers attains an overall gap of on mixed distributions, while plain GNNs on standard benchmarks like CATS do not beat Gurobi, underscoring the value of instance-dependent solver selection. The findings advocate for a hybrid, structure-aware approach where ML identifies hardness and guides the deployment of traditional optimization techniques, achieving high-quality results with limited computational overhead.

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 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.
Paper Structure (29 sections, 3 theorems, 4 equations, 4 figures, 9 tables)

This paper contains 29 sections, 3 theorems, 4 equations, 4 figures, 9 tables.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Winner Determination Problem
  4. Greedy Heuristic and Failure Modes
  5. The Algorithm Selection Problem
  6. Learning Structural Hardness
  7. Feature Design
  8. Classifier Architecture
  9. Results
  10. Adversarial Hardness and GNN Specialist
  11. Whale-Fish Trap Formalization
  12. GNN Specialist Architecture
  13. Trap Variation Results
  14. Hybrid Allocator
  15. Experiments and Analysis
  16. ...and 14 more sections

Key Result

Theorem 1

Consider a WDP instance $\mathcal{I}_k$ with $m$ items and bids $\{b_W, b_{F_1}, \ldots, b_{F_k}\}$ where the whale bid $b_W$ has value $v_W = 1 + \epsilon$ and requests all $m$ items, and each fish bid $b_{F_i}$ has value $v_{F_i} = 1$ and requests a disjoint subset of items with $\bigcup_{i=1}^k \

Figures (4)

  • Figure 1: Overview. (a) The hardness classifier predicts greedy optimality gaps with 0.937 correlation. (b) Whale-fish trap: greedy selects the whale (value 100), missing the collectively superior fish (total 120). (c) Algorithm selection workflow routes hard instances to the GNN and easy instances to greedy, achieving 0.51% overall gap.
  • Figure 2: Hardness classifier analysis. (a) Predicted vs. true greedy gap with regression line ($r$$=$0.937). (b) Classification accuracy vs. threshold for three seeds; the classifier maintains $>$94% accuracy across a wide threshold range.
  • Figure 3: GNN bid-level analysis on trap instances. (a) Probability distributions: the GNN assigns near-zero probability to whale bids and ${\approx}$0.71 to fish bids. (b) Fish-whale probability difference is positive in 100% of instances, confirming systematic fish preference.
  • Figure 4: Distribution of bid density CV for hard (trap) and easy (random) instances. The threshold $\theta = 0.35$ cleanly separates the two populations, enabling 99% routing accuracy.

Theorems & Definitions (4)

  • Theorem 1: Greedy Competitive Ratio on $k$-Star Instances
  • Theorem 2: WDP-MWIS Equivalence
  • proof
  • Corollary 3: Domain Invariance