A Study of Parallel Continuous Local Search

We present Accelerated Fourier SAT (AFSAT), a GPU-accelerated solver for pseudo-Boolean satisfiability based on continuous local search (CLS). AFSAT realises the proof-of-concept approach, FastFourierSAT, into a fully-engineered solver supporting any heterogeneous mixture of symmetric constraint types and lengths within a single problem instance. Using the JAX compiler, AFSAT leverages pure function composition, automatic vectorisation, automatic differentiation, and just-in-time (JIT) compilation to perform massively parallel CLS across batches of candidate assignments. We demonstrate substantially improved numerical stability, runtime performance, and memory efficiency over the proof-of-concept. We achieve this by way of identifying and addressing various limitations that arise from memory latency and floating-point representation, as well as leveraging automatic parallelisation and compact representations. The inherent representational and stability limitations of floating point are partially addressed by a tailored discrete Fourier transform implementation. We achieve near-linear throughput when scaling to multiple accelerators via JAX array sharding.

Optimization problems such as the NP-complete 3-SAT provide an important benchmark for the difficult task of finding ground-states in strongly correlated many-body systems with rugged energy landscapes. The study of random 3-SAT problems as Ising spin Hamiltonians in statistical physics has yielded major insights including the existence of a satisfiability phase transition, and the prediction of a critical parameter line of particularly hard instances. Yet, progress on solving those instances has been scarce for several decades. Here, introducing the Target-SAT (TSAT) algorithm, we roughly triple the tractable problem sizes in the hardest regime, with an even greater improvement in a vast range of neighboring regions. By leveraging statistical information hidden in the combinatorial constraints of the problem, TSAT is actively guided in its stochastic local search toward a target within the relevant parameter space. Our analysis also explains why established local search algorithms are limited to relatively small system sizes due to a vast low-energy trap. Furthermore, we characterize the aforementioned critical line in terms of a dominant additional complexity barrier, whose exponential scaling is quickly overcome by TSAT only in the surrounding parameter space. With TSAT, the lead in solving the hardest known random satisfiability problems returns to the realm of stochastic local search algorithms.

Foundational optimization embeddings have recently emerged as powerful pre-trained representations for mixed-integer programming (MIP) problems. These embeddings were shown to enable cross-domain transfer and reduce reliance on solver-generated labels. In this work, we investigate whether such representations generalize beyond optimization to decision problems, focusing on Boolean satisfiability (SAT). We adapt the foundational optimization architecture to SAT by mapping CNF formulas into the same bipartite constraint-variable graph representation used for MIPs. This allows direct reuse of the pre-trained embedding model without architectural changes or supervised fine-tuning. Our results show that these embeddings capture structural regularities in SAT instances and support unsupervised tasks such as instance clustering and distribution identification. We demonstrate, for the first time, that foundational optimization embeddings can transfer to constraint satisfaction domains. Our findings is a step toward a unified representational framework for both optimization and decision problems.