Learn2Aggregate: Supervised Generation of Chvátal-Gomory Cuts Using Graph Neural Networks

TL;DR

A machine learning framework for optimizing the generation of Chvatal-Gomory (CG) cuts in mixed integer linear programming (MILP) that trains a graph neural network to classify useful constraints for aggregation in CG cut generation, resulting in enhanced CG cut generation across five diverse MILP benchmarks.

Abstract

We present $\textit{Learn2Aggregate}$, a machine learning (ML) framework for optimizing the generation of Chvátal-Gomory (CG) cuts in mixed integer linear programming (MILP). The framework trains a graph neural network to classify useful constraints for aggregation in CG cut generation. The ML-driven CG separator selectively focuses on a small set of impactful constraints, improving runtimes without compromising the strength of the generated cuts. Key to our approach is the formulation of a constraint classification task which favours sparse aggregation of constraints, consistent with empirical findings. This, in conjunction with a careful constraint labeling scheme and a hybrid of deep learning and feature engineering, results in enhanced CG cut generation across five diverse MILP benchmarks. On the largest test sets, our method closes roughly $\textit{twice}$ as much of the integrality gap as the standard CG method while running 40$% faster. This performance improvement is due to our method eliminating 75% of the constraints prior to aggregation.

Figures (8)

• Figure 1: A binary two-variable problem. Constraints are solid and their associated normal vectors (excluding the bound constraints) are drawn, along with the maximization objective vector. Point "LP Opt." is the solution of the LP relaxation whereas "ILP Opt." is the desired integer optimum. The dashed cut, $x_1+x_2\leq 1$ in purple, is a strong cut which we would like to derive by aggregating the (normal vectors of the) dashed constraints; this cut tightens the LP such that its optimum is "ILP Opt.". The normal vector of the desired cut is in the cone of the blue and orange constraints and thus can be derived by aggregating them with appropriate weights (e.g., multiplying both of them with 0.9, adding them up, then rounding down the resulting vector; this is a CG cut). The green constraint is not useful and can be excluded from consideration when aggregating.
• Figure 2: Schematic of the constraint classification GNN with constraint node features $\boldsymbol{C}$, variable node features $\boldsymbol{V}$, and edge features $\boldsymbol{E}$. The embedding size is $h$.
• Figure 3: Mean IGC and standard deviation (shaded) v.s. round for medium/large test instances. The reduced and full CG-MIP separators are shown in red and blue, respectively.
• Figure 4: Plot of final test IGC v.s. percentile of instances. The reduced separator is shown in red, the full separator in blue, GMI cuts in yellow, and 1 round of GMI cuts in green. A larger area under a curve is preferred. Due to space limits, plots for SC and PM datasets are in Appendix \ref{['fig:igc_percentile_plot_PM_SC']}.
• Figure 5: Plot of mean test IGC v.s. cut generation round for full and reduced separators with two labeling strategies.