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Towards a General Framework for Predicting and Explaining the Hardness of Graph-based Combinatorial Optimization Problems using Machine Learning and Association Rule Mining

Bharat Sharman, Elkafi Hassini

TL;DR

This work presents GCO-HPIF, a general framework that uses problem-agnostic graph features to predict the hardness of graph-based COP instances and employs association rule mining to provide interpretable explanations. The MCP case study shows state-of-the-art predictive accuracy (weighted F1 near 0.99 and ROC-AUC around 0.91) with just three features, and ARM yields high-support, interpretable rules. The framework also predicts computation times for GNN-based solvers with high accuracy, while exact solvers remain challenging to model due to their exponential scaling. The approach advances explainable, generalizable hardness prediction and time prediction for COPs, with future work focusing on broader COP coverage, scalability to larger graphs, and refined hardness metrics.

Abstract

This study introduces GCO-HPIF, a general machine-learning-based framework to predict and explain the computational hardness of combinatorial optimization problems that can be represented on graphs. The framework consists of two stages. In the first stage, a dataset is created comprising problem-agnostic graph features and hardness classifications of problem instances. Machine-learning-based classification algorithms are trained to map graph features to hardness categories. In the second stage, the framework explains the predictions using an association rule mining algorithm. Additionally, machine-learning-based regression models are trained to predict algorithmic computation times. The GCO-HPIF framework was applied to a dataset of 3287 maximum clique problem instances compiled from the COLLAB, IMDB, and TWITTER graph datasets using five state-of-the-art algorithms, namely three exact branch-and-bound-based algorithms (Gurobi, CliSAT, and MOMC) and two graph-neural-network-based algorithms (EGN and HGS). The framework demonstrated excellent performance in predicting instance hardness, achieving a weighted F1 score of 0.9921, a minority-class F1 score of 0.878, and an ROC-AUC score of 0.9083 using only three graph features. The best association rule found by the FP-Growth algorithm for explaining the hardness predictions had a support of 0.8829 for hard instances and an overall accuracy of 87.64 percent, underscoring the framework's usefulness for both prediction and explanation. Furthermore, the best-performing regression model for predicting computation times achieved a percentage RMSE of 5.12 and an R2 value of 0.991.

Towards a General Framework for Predicting and Explaining the Hardness of Graph-based Combinatorial Optimization Problems using Machine Learning and Association Rule Mining

TL;DR

This work presents GCO-HPIF, a general framework that uses problem-agnostic graph features to predict the hardness of graph-based COP instances and employs association rule mining to provide interpretable explanations. The MCP case study shows state-of-the-art predictive accuracy (weighted F1 near 0.99 and ROC-AUC around 0.91) with just three features, and ARM yields high-support, interpretable rules. The framework also predicts computation times for GNN-based solvers with high accuracy, while exact solvers remain challenging to model due to their exponential scaling. The approach advances explainable, generalizable hardness prediction and time prediction for COPs, with future work focusing on broader COP coverage, scalability to larger graphs, and refined hardness metrics.

Abstract

This study introduces GCO-HPIF, a general machine-learning-based framework to predict and explain the computational hardness of combinatorial optimization problems that can be represented on graphs. The framework consists of two stages. In the first stage, a dataset is created comprising problem-agnostic graph features and hardness classifications of problem instances. Machine-learning-based classification algorithms are trained to map graph features to hardness categories. In the second stage, the framework explains the predictions using an association rule mining algorithm. Additionally, machine-learning-based regression models are trained to predict algorithmic computation times. The GCO-HPIF framework was applied to a dataset of 3287 maximum clique problem instances compiled from the COLLAB, IMDB, and TWITTER graph datasets using five state-of-the-art algorithms, namely three exact branch-and-bound-based algorithms (Gurobi, CliSAT, and MOMC) and two graph-neural-network-based algorithms (EGN and HGS). The framework demonstrated excellent performance in predicting instance hardness, achieving a weighted F1 score of 0.9921, a minority-class F1 score of 0.878, and an ROC-AUC score of 0.9083 using only three graph features. The best association rule found by the FP-Growth algorithm for explaining the hardness predictions had a support of 0.8829 for hard instances and an overall accuracy of 87.64 percent, underscoring the framework's usefulness for both prediction and explanation. Furthermore, the best-performing regression model for predicting computation times achieved a percentage RMSE of 5.12 and an R2 value of 0.991.
Paper Structure (20 sections, 4 equations, 5 figures, 4 tables)

This paper contains 20 sections, 4 equations, 5 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Graph-based combinatorial optimization problems
  3. Literature review
  4. Significance of generic graph-based features
  5. GCO-HPIF : Framework for predicting and explaining hardness of graph-based COPs
  6. Explaining prediction results using association rule mining
  7. Association rule mining
  8. FP-Growth algorithm
  9. Application of the framework - Maximum Clique Problem
  10. Maximum Clique Problem
  11. Algorithms
  12. Dataset
  13. Hardness measure
  14. Computational considerations
  15. Machine learning based hardness prediction methodology
  16. ...and 5 more sections

Figures (5)

  • Figure 1: Categories of graph-based combinatorial optimization problems
  • Figure 2: GCO-HPIF - a framework for predicting and explaining the hardness of graph-based COPs. *Please refer to Figure \ref{['fig:ml_model_development_instance_hardness_prediction']} that shows the machine learning pipeline for predicting the hardness of problem instances
  • Figure 3: Steps of the machine learning model pipeline for instance hardness prediction
  • Figure 4: Model performance metrics vs number of features for the SVC model
  • Figure 5: Residuals plot of the 658 test instances for RF regression model for HGS computation time prediction