Learning to Rank the Initial Branching Order of SAT Solvers

Arvid Eriksson; Gabriel Poesia; Roman Bresson; Karl Henrik Johansson; David Broman

cs.AIMar 7, 2026

Learning to Rank the Initial Branching Order of SAT Solvers

Authors: Arvid Eriksson, Gabriel Poesia, Roman Bresson, Karl Henrik Johansson, David Broman

Organizations: KTH Royal Institute of Technology · Kempner Institute at Harvard University · Mohamed Bin Zayed University of Artificial Intelligence

Abstract

Finding good branching orders is key to solving SAT problems efficiently, but finding such branching orders is a difficult problem. Using a learning based approach to predict a good branching order before solving, therefore, has potential. In this paper, we investigate predicting branching orders using graph neural networks as a preprocessing step to conflict-driven clause learning (CDCL) SAT solvers. We show that there are significant gains to be made in existing CDCL SAT solvers by providing a good initial branching. Further, we provide three labeling methods to find such initial branching orders in a tractable way. Finally, we train a graph neural network to predict these branching orders and show through our evaluations that a GNN-initialized ordering yields significant speedups on random 3-CNF and pseudo-industrial benchmarks, with generalization capabilities to instances much larger than the training set. However, we also find that the predictions fail at speeding up more difficult and industrial instances. We attribute this to the solver's dynamic heuristics, which rapidly overwrite the provided initialization, and to the complexity of these instances, making GNN prediction hard.

Figures & tables

Panel of Figure 1 — Figure 1: Pipeline for training (top) and inference (bottom) for a branch order-predicting GNN preprocessing step. The orange boxes (dotted outline) signify steps that use a modified SAT solver, while purple boxes (dashed outline) signify steps using a graph neural network.

Figure 2: Mean solving time for setting one of 50 randomly sampled variables first, with the remaining order randomized. The shaded areas indicate a 95% confidence interval over 1000 iterations for each variable.

Instance	Variables	Clauses	Median (s)	Best speedup (%)	Top 10% speedup (%)
3-CNF 100	$100$	$430$	$0.0007$	$2233.3$	$536.4$
3-CNF 150	$150$	$645$	$0.016$	$63.1$	$30.8$
3-CNF 200	$200$	$860$	$0.12$	$13.1$	$9.1$
3-CNF 250	$250$	$1075$	$0.98$	$5.6$	$4.1$
Ramsey	$37\,410$	$162\,400$	$3.41$	$0.8$	$0.5$
Manthey	$3996$	$29\,400$	$0.59$	$2.0$	$0.9$

Table 1: Solving time and speedup when choosing the best and 10th-percentile variables, respectively, compared to the median, when forcing initial branching to start on a specific variable.

Figure 3: Graph of

(x_{1}\vee\neg x_{2})\wedge(x_{2}\vee x_{3})

Figure 4: Mean reduction in the number of propagations relative to the default variable order on SATLIB uniform random 3-CNF benchmarks. Top: CaDiCaL on SAT and UNSAT instances, respectively. Bottom: MiniSat on SAT and UNSAT instances, respectively. Shaded regions denote a 95% confidence interval across the dataset.

Figure 5: Number of propagations on the hard G4SATBench instances. Speedups of 10%-20% until roughly

10^{5}

propagations.

Figure 6: Solving time for CaDiCaL and MiniSat on the SAT competition 2024 main track.

Appendix figures & tables4 assets

Supplementary material from the paper’s appendix.

Appendix

Panel of Figure 7 — Figure 7: Comparison of inference time and solving time using CaDiCaL (default order) when solving on CPU only for SAT (a) and UNSAT (b) instances. Note the logarithmic y-axis.

Hyperparameter	Value
Model Architecture	Graph Transformer from NeuroBack ( Wang et al., 2024 )
Optimizer	AdamW ( Loshchilov and Hutter, 2018 )
Adam $\beta_{1}$	0.9
Adam $\beta_{2}$	0.999
Learning Rate	0.001
Epochs	10

Appendix

Table 2: Model hyperparameters

Figure 8: Number of propagations on the hard G4SATBench instances. No distinct speedup.

Figure 9: Number of propagations for different reminding factors. Only the problems that all seven initializations solve are plotted.

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Learning-based approaches to NP-hard problems have shown increasing promise, but their progress is fundamentally constrained by the high cost of generating labeled training data. In domains such as Boolean satisfiability (SAT), standard pipelines rely on solver-in-the-loop labeling, which scales poorly with problem size and limits the amount of usable supervision. This bottleneck hinders the broader goal of leveraging machine learning to capture structure in hard combinatorial problems. In this work, we propose a target-aware, solver-free data generation framework for SAT that produces correctly labeled SAT and UNSAT instances by construction, eliminating the need for expensive solver calls. Our method aligns generated instances with the structural properties of a target benchmark, making synthetic data effective for downstream learning. We further develop a linear-programming-aware graph neural network (LPGNN) architecture that incorporates constraint-violation residuals into message passing, enabling the model to exploit underlying optimization structure. Together, these contributions support a data-centric paradigm for learning on NP-hard problems, where scalable, task-aligned data generation is as critical as model design. Our approach yields orders-of-magnitude speedups in data generation, demonstrating that benchmark-aligned synthetic data can effectively augment solver-labeled datasets for GNN-based SAT prediction.

Abstract

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