Title: Graph-Based Attention for Differentiable MaxSAT Solving
Venue: NeurIPS 2025
Accepted: yes
Keywords: Maximum Satisfiability

Weighted Maximum Satisfiability

Graph Neural Network

Graph Attention

T-Norm

Neurosymbolic
Abstract: The use of deep learning to solve fundamental AI problems such as Boolean Satisfiability (SAT) has been explored recently to develop robust and scalable reasoning systems. This work advances such neural-based reasoning approaches by developing a new Graph Neural Network (GNN) to differentiably solve (weighted) Maximum Satisfiability (MaxSAT). To this end, we propose SAT-based Graph Attention Networks (SGATs) as novel GNNs that are built on t-norm based attention and message passing mechanisms, and structurally designed to approximate greedy distributed local search. To demonstrate the effectiveness of our model, we develop a local search solver that uses SGATs to continuously solve any given MaxSAT problem. Experiments on (weighted) MaxSAT benchmark datasets demonstrate that SGATs significantly outperform existing neural-based architectures, and achieve state-of-the-art performance among continuous approaches, highlighting the strength of the proposed model.


Peer reviews:
comments: SUMMARY:
This paper presents SAT-based Graph Attention Networks (SGATs), a novel class of Graph Neural Networks (GNNs) designed to solve (weighted) MaxSAT problems via continuous optimization. SGATs integrate t-norm based attention and message-passing mechanisms to approximate greedy distributed local search. A local search solver, LS-GNN, is developed to leverage SGATs iteratively.

STRENGTHS AND WEAKNESSES:
Strengths:

+ The local search framework LS-GNN is simple yet effective, and fits naturally with the design of SGATs.
+ The paper includes ablation studies and theoretical analysis (approximation ratio) that enhance its technical credibility.

Weaknesses:

+ The continuous optimization baselines (Mixing from 2019, MIXSAT from 2017, FourierSAT from 2020) are relatively outdated. The paper would be more convincing if compared against more recent local search-based MaxSAT solvers.
+ While the proposed SGAT architecture is novel, the motivation for using GNNs over conventional solvers could be made more explicit.

QUESTIONS:
1. Why were the ontinuous optimization baselines restricted to pre-2020 methods? Have more recent MaxSAT solvers been considered?

2. While the ablation studies on MS2018 are informative, could the authors clarify whether similar analyses were attempted on more recent benchmarks? It would help understand whether the observed component-wise benefits generalize beyond the training domain.

LIMITATIONS:
Yes.
TITLE: Official Review
CLARITY: 3
IMPACT: 3
ORIGINALITY: 3
DATE: 2025-06-20

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposes SGAT, a novel Graph Neural Network architecture, designed specifically to address (Weighted) MaxSAT problems. SGAT incorporates a t-norm-based attention and message passing mechanism that approximates greedy distributed local search in a differentiable framework. The authors also introduce LS-GNN, a local search solver leveraging SGATs. The proposed method significantly outperforms existing GNN architectures and continuous optimisation baselines such as Mixing, MIXSAT, and FourierSAT on multiple MaxSAT benchmarks. Additional experiments demonstrate the importance of SGAT's components, and highlight its potential to provide high-quality initial assignments for integration with existing SOTA solvers.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths\n---\n- The definitions of the target problems (SAT, MaxSAT, Weighted MaxSAT) are clearly and precisely explained. The mathematical formulations provided throughout the paper are well-defined, making the paper highly readable and accessible.\n- The proposed GAT-based architecture, SGAT, is systematically well-structured and clearly explained. The roles of all variables in the equations are explicitly defined, and the message passing mechanism and optimisation objectives are well-formulated.\n- SGAT significantly outperforms baseline GNN architectures such as NeuroSAT, GGNN, and MS-ESFG in experimental results. Moreover, the LS-GNN algorithm, built on SGAT, exhibits superior performance compared to continuous MaxSAT solver baselines like Mixing, MIXSAT, and FourierSAT.\n- A well-constructed ablation study clearly shows that each component of SGAT (SGAT Layer, T-norm Layer, T-norm type) plays a crucial role in its high performance.\n- SGAT demonstrates practical utility when used as an initialisation heuristic for existing SOTA Weighted MaxSAT solvers. The experimental report suggests that integrating SGAT can improve solver performance, indicating the high potential of SGAT in real-world solver integration.\n\nWeaknesses\n---\n- In Algorithm 1 (LS-GNN), it is unclear how $\\Delta$ in Step 18 is defined. A brief explanation of \"early stopping\" variable would be appreciated. In the while loop, $\\alpha$ is repeatedly re-initialised with a random assignment in Step 4, and it remains unsure to me how the update in Step 18 is contributing to the process.\n- In Lines 224–225, the paper mentions an analysis of the approximation ratio of SGATs. It would be also appreciated to provide some concrete numbers or at least a summary (better or worse than prior methods) regarding this ratio, even if full theoretical guarantees are difficult.\n\nQUESTIONS:\n- In Algorithm 1, is the local search within the while loop executed in parallel? Is it possible to assign multiple random assignments to $\\alpha$ simultaneously, conduct local search in parallel, and then choose the one with the best cost? If the process executes with a single $\\alpha$, is parallelisation feasible, and could it lead to performance improvements?\n- In Figure 4, the test performance is generally better when models are trained on the synthetic dataset SR(U(40,200)) rather than the benchmark MS2018(2MB). This seems counter-intuitive. Could the authors provide an explanation for this phenomenon?\n- In the formula for Average Clause Satisfaction (Line 244), why is the round function applied to the clause satisfaction value? It seems possible to compute a meaningful average without rounding. What is the rationale for this choice?\n- In Table 1, some baseline solvers perform worse under a 300-second timeout than under a 60-second timeout in certain years. Could the authors explain why longer time limits would degrade performance in these cases?\n\nLIMITATIONS:\nThe main limitations of this work are those already acknowledged by the authors. First, SGAT has not yet been extended to handle partial MaxSAT problems, where satisfying hard clauses is mandatory. Second, while the method shows strong performance on standard MaxSAT benchmarks, it has not been tested in broader neuro-symbolic reasoning tasks or real-world domains beyond SAT-style formulations. These limitations are rather natural next steps for future exploration. Given the flexibility of the proposed architecture and its compatibility with existing solvers, these extensions seem feasible and promising.", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 4, "CLARITY": 4, "IMPACT": 4, "ORIGINALITY": 3, "DATE": "2025-06-26"}

comments: SUMMARY:
The paper proposes a solver based on a graph neural network. The authors present the architecture and how the model is the use for solving MaxSAT problems. 

After the model and algorithm presentation the authors show the performance on standard MaxSAT instances and how the proposed approach works with ablation study on architecture choices.

STRENGTHS AND WEAKNESSES:
Strengths 
1. clear presentation of the architecture and the separation of the role of the network and the algorithm
2. Experiments that highlight the performance and the role of the architecture choices.  

I personally like that the bipartite graph, the transformation from variable to clause is base on the T-norm. 

Weaknesses
1. I would have like to see the analysis with other heuristics

QUESTIONS:
It isr elated to the weakeness point: would it be possible to define a different local search? Could you elaborate? there is part in the ANNEX that extend, could you spring something in the main text?


In algorithm 1 is not clear what \alpha refers to, since there is the \alpha of the attention, but this is not a scalar

In general, it is not clear what is learned and how it is learned. 

It would also be nice to see where the approach is with respect to more mature solvers: "MaxSAT Evaluation 2023: Summary of Unweighted Exact Track"
(https://maxsat-evaluations.github.io/2023/results/exact/unweighted/summary.html) or any other source.

LIMITATIONS:
yes, in the annex
TITLE: Official Review
CLARITY: 3
IMPACT: 3
ORIGINALITY: 3
DATE: 2025-07-01

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe authors propose SGAT, a graph attention network architecture specifically designed to solve the MaxSAT problem, which is claimed to be the first GNNs to handle weighted MaxSAT problems. The technical contributions include the t-norm based attention and SGAT layer, an approximation of greedy distributed local search. SGAT is demonstrated to achieve better solutions compared with existing GNN models on MaxSAT problems from both randomly generated and competition benchmarks. Moreover, by integrating SGAT with a local search framework, the proposed algorithm outperforms other continuous optimisation algorithms in runtime on weighted MaxSAT Evaluation benchmarks.\n\nSTRENGTHS AND WEAKNESSES:\n### Strengths\n- The proposed SGAT is novel and efficient, which is the first GNN model to work on weighted MaxSAT instances, a critical problem of various domains in CS&AI.\n- The t-norm attention and SGAT layer are well-motivated, and ablation studies have demonstrated their effectiveness and stability.\n- Experiments on MaxSAT Evaluation benchmarks are promising, which have shown the ability of NN-based approach on these challenging practical problems.\n- The manuscript is generally well-organised and easy to follow.\n\n### Weaknesses\n- Readers may find the term \"t-norm attention\" confusing, as it is not explained until Section 4.2. Explaining it when first mentioned would improve the clarity.\n- It should be clarified that Algorithm 1 can solve weighted MaxSAT instances to strengthen the contribution.\n- There is no meaningful conclusion from Section 4.6.\n\nQUESTIONS:\n- Has a comparison been made with traditional discrete search solvers? If so, does SGAT-based local search have any advantages?\n- How was the choice made to have 6 SGAT blocks in the model, and what effects would setting more or fewer layers have on the results?\n- According to my knowledge, some weighted MaxSAT benchmarks in MaxSAT Evaluations have more than 100,000 variables. How can such large graphs be loaded onto the GPUs for training?\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-01"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper introduces a new neural network for solving the (weighted) MaxSAT problem. Having converted a MaxSAT problem to an appropriate graph representation, the architecture consists of 3 parts: (i) T-norm Aggregation, (ii) SGAT layers, (iii) A normalization layer. The model is then be used in a local search framework to complete the approach. The authors include ablations to confirm their design choice of using SGAT (vs GAT) and the T-norm and for testing different T-norms. The authors provide results on two MaxSAT datasets, comparing the approach with previous SAT or MaxSAT architectures. Their model produces favorable results.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths\n* Favorable results when compared with existing neural network SAT solvers \n* Nice ablations\n\nWeaknesses\n* Contribution: Like several other papers in this direction of research (combinatorial optimization with neural networks), I feel this work does not convince me that using neural networks is necessary or even helpful here. The proposed approach aligns very closely with greedy distributed local search (as clearly stated by the authors - which I appreciate), and I am not sure what it contributes on top:\n* So firstly: CBLM and an appropriate greedy local search should be included as a baseline. The proposed model should outperform these baselines to justify itself. \n* Secondly: As far as I can tell with this approach there is not much transfer of knowledge between training and testing. Each problem instance essentially has to be solved separately. Perhaps the authors could elaborate on what knowledge they expect the model weights to be able to learn which can be transferred between problem instances. I would appreciate some intuition here to motivate the approach. To stress this point further, the update messages passed between nodes represent the value in which the clause wants the variable to be updated, rather than some more complex embeddings that somehow represent the structure/topology of the problem instance, which could be transferable between problem instances. I'm happy to receive pushback on this if the authors do not agree with my intuition. \n\nQuestions\n* Is the average clause satisfaction the standard metric here? I would have expected that the variables would be rounded (binarized) first, and then the (weighted) number of satisfied clauses calculated.\n\nQUESTIONS:\nPlease refer to the strengths and weaknesses.\n\nLIMITATIONS:\nI think the limitations could be discussed in more detail - please see weaknesses.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-07-02"}