Abstract. The research community on complex networks has developed techniques of analysis and algorithms that can be used by the SAT community to improve our knowledge about the structure of industrial SAT instances. It is often argued that modern SAT solvers are able to exploit this hidden structure, without a precise definition of this notion. In this paper, we show that most industrial SAT instances have a high modularity that is not present in random instances. We also show that successful techniques, like learning, (indirectly) take into account this community structure. Our experimental study reveal that most learnt clauses are local on one of those modules or communities.

Despite the success of CDCL SAT solvers solving industrial problems, there are still many open questions to explain such success. In this context, the generation of random SAT instances having computational properties more similar to realworld problems becomes crucial. Such generators are possibly the best tool to analyze families of instances and solvers behaviors on them. In this paper, we present a random SAT instances generator based on the notion of locality. We show that this is a decisive dimension of attractiveness among the variables of a formula, and how CDCL SAT solvers take advantage of it. To the best of our knowledge, this is the first random SAT model that generates both scale-free structure and community structure at once.

Nowadays, modern SAT solvers are able to efficiently solve many industrial, or realworld, SAT instances. However, the process of development and testing of new SAT solving techniques is conditioned to the finite and reduced number of known industrial benchmarks. Therefore, new models of random SAT instances generation that capture realistically the features of real-world problems can be beneficial to the SAT community. In many works, the structure of industrial instances has been analyzed representing them as graphs and studying some of their properties, like modularity. In this work, we use the notion of modularity to define a new model of generation of random SAT instances with community structure, called Community Attachment. For high values of modularity (i.e., clear community structure), we realistically model pseudoindustrial random SAT formulas. This model also generates SAT instances very similar to classical random formulas using a low value of modularity. We also prove that the phase transition point, if exists, is independent on the modularity. We evaluate the adequacy of this model to real industrial SAT problems in terms of SAT solvers performance, and show that modern solvers do actually exploit this community structure. Finally, we use this generator to observe the connections between the modularity of the instance and some components of the solver, such as the variable branching heuristics or the clause learning mechanism.

Nowadays, Conflict-Driven Clause Learning (CDCL) techniques are one of the key components of modern SAT solvers specialized in industrial instances. Last years, one of the focuses has been put on strategies to select which learnt clauses are removed during the search. Originally, one need for removing clauses was motivated by the finiteness of memory. Recently, it has been shown that more aggressive clause deletion policies may improve solvers performance, even when memory is sufficient. Also, the utility of learnt clauses has been related to the modular structure of industrial SAT instances. In this paper, we show that augmenting SAT instances with learnt clauses does not always make them easier for the SAT solver. In fact, it makes worse the solver performance in many cases. However, we identify a set of highly useful learnt clauses, and we show that augmenting SAT instances with this set of clauses contributes to improve the solver performance in many cases, especially in satisfiable formulas. These clauses are related to the community structure of the formula, and they can be computed in a fast preprocessing step. This would suggest that the community structure may play an important role in clause deletion policies.