In the last decades, we have witnessed a remarkable success of algorithms solving the Boolean Satisfiability problem (SAT) on instances encoding application or real-world problems arising from a very diverse number of domains, such as hardware and software verification, planning or cryptography. These algorithms are the so known Conflict-Driven Clause Learning (CDCL) SAT solvers. Interestingly enough, the reasons for the success of these solvers on this diverse range of problems are not completely understood yet.

A common issue when facing this open challenge is the heterogeneity of this set of benchmarks. Another problem is the limited number of existing instances. In this context, random models of SAT formulas capturing features shared by the majority of these application benchmarks become crucial, for both theoretical and practical purposes. On the one hand, it is undoubtedly necessary to have random models where theoretical properties, like hardness, can be studied. Therefore, realistic random SAT models may contribute to explain the success of these solvers on these industrial problems. On the other hand, the limited number of benchmarks and their hardness in practice makes the evaluation of new solving techniques a costly task. Therefore, these realistic random SAT generators can provide an unlimited number of pseudo-industrial random SAT instances with some desired properties.

In this work, we present a random SAT instances generator based on the notion of locality. This notion is complementary to the popularity of variables, which is present in the scale-free structure, observable in actual application problems and achievable by previous generators. Our random SAT model combines both locality and popularity, and we show that they are two decisive dimensions of attractiveness among the variables of a formula, and how CDCL SAT solvers take advantage of them. Locality is closely related to the community structure, another important feature of application SAT benchmarks, which is indirectly achieved by this model. To the best of our knowledge, this is the first random SAT model that generates both scale-free structure and community structure at once.

在过去的几十年里，我们目睹了算法在解决布尔可满足性问题 (SAT) 的实例编码应用程序或现实世界问题上取得的巨大成功，这些问题来自于硬件和软件验证、规划或密码学等非常多样化的领域。这些算法是众所周知的冲突驱动子句学习 (CDCL) SAT 求解器。有趣的是，这些求解器在这些不同范围的问题上取得成功的原因尚不完全清楚。

面对这一公开挑战时，一个常见问题是这组基准的异质性。另一个问题是现有实例的数量有限。在这种情况下，对于理论和实践而言，捕获大多数这些应用程序基准共享的特征的 SAT 公式随机模型变得至关重要。一方面，毫无疑问有必要拥有可以研究硬度等理论特性的随机模型。因此，现实的随机 SAT 模型可能有助于解释这些求解器在这些工业问题上的成功。另一方面，有限数量的基准及其在实践中的难度使得评估新的求解技术成为一项昂贵的任务。因此，这些现实随机 SAT 生成器可以提供无限数量的具有某些所需属性的伪工业随机 SAT 实例。

在这项工作中，我们提出了一个基于局部性概念的随机 SAT 实例生成器。这个概念是对变量的流行的补充，变量存在于无标度结构中，在实际应用问题中可以观察到，并且可以由以前的生成器实现。我们的随机 SAT 模型结合了局部性和流行性，我们表明它们是公式变量中吸引力的两个决定性维度，以及 CDCL SAT 求解器如何利用它们。局部性与社区结构密切相关，这是应用 SAT 基准测试的另一个重要特征，通过该模型间接实现。据我们所知，这是第一个同时生成无标度结构和社区结构的随机 SAT 模型。