Wepropose SplitGNN, a graph neural network (GNN)-based approach that learns to solve weighted maximum satisfiabil ity (MaxSAT) problem. SplitGNN incorporates a co-training architecture consisting of supervised message passing mech anism and unsupervised solution boosting layer. A new graph representation called edge-splitting factor graph is proposed to provide more structural information for learning, which is based on spanning tree generation and edge classification. To improve the solutions on challenging and weighted instances, we implement a GPU-accelerated layer applying efficient score calculation and relaxation-based optimization. Exper iments show that SplitGNN achieves 3* faster convergence and better predictions compared with other GNN-based ar chitectures. More notably, SplitGNN successfully finds solu tions that outperform modern heuristic MaxSAT solvers on much larger and harder weighted MaxSAT benchmarks, and demonstrates exceptional generalization abilities on diverse structural instances.

We propose a generative, end-to-end solver for black-box combinatorial optimization that emphasizes both sample efficiency and solution quality on NP problems. Drawing inspiration from annealing-based algorithms, we treat the black-box objective as an energy function and train a neural network to model the associated Boltzmann distribution. By conditioning on temperature, the network captures a continuum of distributions--from near-uniform at high temperatures to sharply peaked around global optima at low temperatures--thereby learning the structure of the energy landscape and facilitating global optimization. When queries are expensive, the temperature-dependent distributions naturally enable data augmentation and improve sample efficiency. When queries are cheap but the problem remains hard, the model learns implicit variable interactions, effectively "opening" the black box. We validate our approach on challenging combinatorial tasks under both limited and unlimited query budgets, showing competitive performance against state-of-the-art black-box optimizers.

We introduce and benchmark a stochastic local search heuristic for the NP-complete satisfiability problem 3-SAT that drastically outperforms existing solvers in the notoriously difficult realm of critically hard instances. Our construction is based on the crucial observation that well established previous approaches such as WalkSAT are prone to get stuck in local minima that are distinguished from true solutions by a larger number of oversatisfied combinatorial constraints. To address this issue, the proposed algorithm, coined DOCSAT, dissipates oversatisfied constraints (DOC), i.e. reduces their unfavorable abundance so as to render them critical. We analyze and benchmark our algorithm on a randomly generated sample of hard but satisfiable 3-SAT instances with varying problem sizes up to N=15000. Quite remarkably, we find that DOCSAT outperforms both WalkSAT and other well known algorithms including the complete solver Kissat, even when comparing its ability to solve the hardest quintile of the sample to the average performance of its competitors. The essence of DOCSAT may be seen as a way of harnessing statistical structure beyond the primary cost function of a combinatorial problem to avoid or escape local minima traps in stochastic local search, which opens avenues for generalization to other optimization problems.

Reasoning is a cognitive ability which allows humans to solve problems with previously unseen combinations of constraints. For a long time, it has been debated whether artificial neural networks can obtain such generalization skills or whether they can only learn to detect superficial patterns Fodor and Pylyshyn [1988], Marcus [2003, 2018] without being able to generalize to novel combinations of constraints. With the arrival of Large Language Models (LLMs) specially trained for reasoning Guo et al. [2025], Jaech et al. [2024], it became harder and harder to claim that these models can only detect superficial patterns. Nevertheless, the exact mechanism by which they are able to solve tasks that typically require reasoning is largely unknown and the robustness of the solving process is also not understood. In this contribution, we focus on a restricted class of problems that require reasoning, concretely on solving Boolean formulas in CNF form. This could be viewed as a prototypical task where the goal is to solve problems with novel combinations of constraints, and where detecting superficial patterns seen during training would be insufficient. It has already been demonstrated that Graph Neural Networks (GNNs) can successfully learn to solve such problems and generalize to larger problems Selsam et al. [2018], even though they are still not competitive when compared to state of the art SAT solvers. Understanding the underlying mechanisms GNNs employ to successfully solve problems, as well as their limitations, would offer significant practical and theoretical value.

The remarkable achievements of machine learning techniques in analyzing discrete structures have drawn significant attention towards their integration into combinatorial optimization algorithms. Typically, these methodologies improve existing solvers by injecting learned models within the solving loop to enhance the efficiency of the search process. In this work, we derive a single differentiable function capable of approximating solutions for the Maximum Satisfiability Problem (MaxSAT). Then, we present a novel neural network architecture to model our differentiable function, and progressively solve MaxSAT using backpropagation. This approach eliminates the need for labeled data or a neural network training phase, as the training process functions as the solving algorithm. Additionally, we leverage the computational power of GPUs to accelerate these computations. Experimental results on challenging MaxSAT instances show that our proposed methodology outperforms two existing MaxSAT solvers, and is on par with another in terms of solution cost, without necessitating any training or access to an underlying SAT solver. Given that numerous NP-hard problems can be reduced to MaxSAT, our novel technique paves the way for a new generation of solvers poised to benefit from neural network GPU acceleration.

This paper addresses the resolution of the 3-SAT problem using a QAOA-like approach. The chosen principle involves modeling the solution ranks of the 3-SAT problem, which, in this particular case, directly represent a solution. This results in a highly compact circuit with few gates, enabling the modeling of large-sized 3-SAT problems. Numerical experimentation demonstrates that the approach can solve instances composed of 91 clauses and 20 variables with an implementation based on Qiskit.

The Boolean Satisfiability problem (SAT) is important on artificial intelligence community and the impact of its solving on complex problems. Recently, great breakthroughs have been made respectively on stochastic local search (SLS) algorithms for uniform random k-SAT resulting in several state-of-the-art SLS algorithms Score2SAT, YalSAT, ProbSAT, CScoreSAT and on a hybrid algorithm for hard random SAT (HRS) resulting in one state-of-the-art hybrid algorithm SparrowToRiss. However, there is no an algorithm which can effectively solve both uniform random k-SAT and HRS. In this paper, we present a new SLS algorithm named SelectNTS for uniform random k-SAT and HRS. SelectNTS is an improved probability selecting based local search algorithm for SAT problem. The core of SelectNTS relies on new clause and variable selection heuristics. The new clause selection heuristic uses a new clause weighting scheme and a biased random walk. The new variable selection heuristic uses a probability selecting strategy with the variation of CC strategy based on a new variable weighting scheme. Extensive experimental results on the well-known random benchmarks instances from the SAT Competitions in 2017 and 2018, and on randomly generated problems, show that our algorithm outperforms state-of-the-art random SAT algorithms, and our SelectNTS can effectively solve both uniform random k-SAT and HRS.

We introduce the reinforcement quantum annealing (RQA) scheme in which an intelligent agent interacts with a quantum annealer that plays the stochastic environment role of learning automata and tries to iteratively find better Ising Hamiltonians for the given problem of interest. As a proof-of-concept, we propose a novel approach for reducing the NP-complete problem of Boolean satisfiability (SAT) to minimizing Ising Hamiltonians and show how to apply the RQA for increasing the probability of finding the global optimum. Our experimental results on two different benchmark SAT problems (namely factoring pseudo-prime numbers and random SAT with phase transitions), using a D-Wave 2000Q quantum processor, demonstrated that RQA finds notably better solutions with fewer samples, compared to state-of-the-art techniques in the realm of quantum annealing.

It is feasible and practically-valuable to bridge the characteristics between graph neural networks (GNNs) and logical reasoning. Despite considerable efforts and successes witnessed in learning Boolean satisfiability (SAT), it remains an open question of learning GNN-based solvers for more complex predicate logic formulae. In this work, we conjectures with theoretically support discussion, that generally defined GNNs present some limitations in reasoning about a set of assignments and proving the unsatisfiability (UNSAT) in Boolean formulae. It implies that GNNs may probably fail in learning the logical reasoning tasks if they contain UNSAT as the sub-problem, thus, included by most of predicate logic reasoning problems.

We introduce a new local search algorithm for satisfiability problems. Usual approaches focus uniformly on unsatisfied clauses. The new method works by picking uniformly random variables in unsatisfied clauses. A Variable-based Focused Metropolis Search (V-FMS) is then applied to random 3-SAT. We show that it is quite comparable in performance to the clause-based FMS. Consequences for algorithmic design are discussed.