A I ] 3 J ul 2 01 6 Community Structure in Industrial SAT Instances

Modern SAT solvers have experienced a remarkable progress on solving industrial instances. Most of the techniques have been developed after an intensive experimental process. It is believed that these techniques exploit the underlying structure of industrial instances. However, there are few works trying to exactly characterize the main features of this structure. The research community on complex networks has developed techniques of analysis and algorithms to study real-world graphs that can be used by the SAT community. Recently, there have been some attempts to analyze the structure of industrial SAT instances in terms of complex networks, with the aim of explaining the success of SAT solving techniques, and possibly improving them. In this paper, inspired by the results on complex networks, we study the community structure, or modularity, of industrial SAT instances. In a graph with clear community structure, or high modularity, we can find a partition of its nodes into communities such that most edges connect variables of the same community. In our analysis, we represent SAT instances as graphs, and we show that most application benchmarks are characterized by a high modularity. On the contrary, random SAT instances are closer to the classical Erdös-Rényi random graph model, where no structure can be observed. We also analyze how this structure evolves by the effects of the execution of the SAT solver. We detect that new clauses learnt by the solver during the search contribute to destroy the original community structure of the formula. This partially explains the distinct performance of SAT solvers on random and industrial SAT instances.

[1]  Jordi Levy,et al.  Generating SAT instances with community structure , 2016, Artif. Intell..

[2]  Carlos Ansótegui,et al.  Using Community Structure to Detect Relevant Learnt Clauses , 2015, SAT.

[3]  Krzysztof Czarnecki,et al.  SATGraf: Visualizing the Evolution of SAT Formula Structure in Solvers , 2015, SAT.

[4]  Mikolás Janota,et al.  Exploiting Resolution-Based Representations for MaxSAT Solving , 2015, SAT.

[5]  Laurent Simon,et al.  Post Mortem Analysis of SAT Solver Proofs , 2014, POS@SAT.

[6]  Tomohiro Sonobe,et al.  Community Branching for Parallel Portfolio SAT Solvers , 2014, SAT.

[7]  Sebastian Fischmeister,et al.  Impact of Community Structure on SAT Solver Performance , 2014, SAT.

[8]  Maria Luisa Bonet,et al.  The Fractal Dimension of SAT Formulas , 2013, IJCAR.

[9]  Vasco M. Manquinho,et al.  Community-Based Partitioning for MaxSAT Solving , 2013, SAT.

[10]  George Katsirelos,et al.  Eigenvector Centrality in Industrial SAT Instances , 2012, CP.

[11]  Carlos Ansótegui,et al.  The Community Structure of SAT Formulas , 2012, SAT.

[12]  Michael Kaufmann,et al.  Creating Industrial-Like SAT Instances by Clustering and Reconstruction - (Poster Presentation) , 2012, SAT.

[13]  Joao Marques-Silva,et al.  Empirical Study of the Anatomy of Modern Sat Solvers , 2011, SAT.

[14]  Maria Luisa Bonet,et al.  On the Structure of Industrial SAT Instances , 2009, CP.

[15]  Gilles Audemard,et al.  Predicting Learnt Clauses Quality in Modern SAT Solvers , 2009, IJCAI.

[16]  Santo Fortunato,et al.  Community detection in graphs , 2009, ArXiv.

[17]  Jean-Loup Guillaume,et al.  Fast unfolding of communities in large networks , 2008, 0803.0476.

[18]  Ulrik Brandes,et al.  On Modularity Clustering , 2008, IEEE Transactions on Knowledge and Data Engineering.

[19]  Réka Albert,et al.  Near linear time algorithm to detect community structures in large-scale networks. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[20]  M. Barber.  Modularity and community detection in bipartite networks. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[21]  Bart Selman,et al.  The state of SAT , 2007, Discret. Appl. Math..

[22]  Andrzej Rucinski,et al.  Random graphs , 2006, SODA.

[23]  Armin Biere,et al.  Effective Preprocessing in SAT Through Variable and Clause Elimination , 2005, SAT.

[24]  M. Newman,et al.  Finding community structure in very large networks. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[25]  Bart Selman,et al.  Ten Challenges Redux: Recent Progress in Propositional Reasoning and Search , 2003, CP.

[26]  M. Newman.  Fast algorithm for detecting community structure in networks. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[27]  M. Newman,et al.  Finding and evaluating community structure in networks. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[28]  Bart Selman,et al.  Backdoors To Typical Case Complexity , 2003, IJCAI.

[29]  Niklas Sörensson,et al.  An Extensible SAT-solver , 2003, SAT.

[30]  Andrew Slater,et al.  Modelling More Realistic SAT Problems , 2002, Australian Joint Conference on Artificial Intelligence.

[31]  Toby Walsh,et al.  Search on High Degree Graphs , 2001, IJCAI.

[32]  Bart Selman,et al.  Satisfiability testing: recent developments and challenge problems , 2000, Proceedings Fifteenth Annual IEEE Symposium on Logic in Computer Science (Cat. No.99CB36332).

[33]  Toby Walsh,et al.  Search in a Small World , 1999, IJCAI.

[34]  Toby Walsh,et al.  Morphing: Combining Structure and Randomness , 1999, AAAI/IAAI.

[35]  Duncan J. Watts,et al.  Collective dynamics of ‘small-world’ networks , 1998, Nature.

[36]  Henry A. Kautz,et al.  Ten Challenges in Propositional Reasoning and Search , 1997, IJCAI.

[37]  Bart Selman,et al.  Problem Structure in the Presence of Perturbations , 1997, AAAI/IAAI.

[38]  Tad Hogg,et al.  Refining the Phase Transition in Combinatorial Search , 1996, Artif. Intell..

[39]  Jesús Giráldez-Cru,et al.  A Modularity-Based Random SAT Instances Generator , 2015, IJCAI.

[40]  Roman Barták,et al.  Constraint Processing , 2009, Encyclopedia of Artificial Intelligence.

[41]  Ilkka Niemelä,et al.  The effect of structural branching on the efficiency of clause learning SAT solving: An experimental study , 2008, J. Algorithms.

[42]  Armin Biere,et al.  Decomposing SAT Problems into Connected Components , 2006, J. Satisf. Boolean Model. Comput..