Impact of Community Structure on SAT Solver Performance

  • Zack Newsham
  • Vijay Ganesh
  • Sebastian Fischmeister
  • Gilles Audemard
  • Laurent Simon
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8561)


Modern CDCL SAT solvers routinely solve very large industrial SAT instances in relatively short periods of time. It is clear that these solvers somehow exploit the structure of real-world instances. However, to-date there have been few results that precisely characterise this structure. In this paper, we provide evidence that the community structure of real-world SAT instances is correlated with the running time of CDCL SAT solvers. It has been known for some time that real-world SAT instances, viewed as graphs, have natural communities in them. A community is a sub-graph of the graph of a SAT instance, such that this sub-graph has more internal edges than outgoing to the rest of the graph. The community structure of a graph is often characterised by a quality metric called Q. Intuitively, a graph with high-quality community structure (high Q) is easily separable into smaller communities, while the one with low Q is not. We provide three results based on empirical data which show that community structure of real-world industrial instances is a better predictor of the running time of CDCL solvers than other commonly considered factors such as variables and clauses. First, we show that there is a strong correlation between the Q value and Literal Block Distance metric of quality of conflict clauses used in clause-deletion policies in Glucose-like solvers. Second, using regression analysis, we show that the the number of communities and the Q value of the graph of real-world SAT instances is more predictive of the running time of CDCL solvers than traditional metrics like number of variables or clauses. Finally, we show that randomly-generated SAT instances with 0.05 ≤ Q ≤ 0.13 are dramatically harder to solve for CDCL solvers than otherwise.


Community Structure Input Formula Current Partial Assignment CDCL Solver Average Solution Time 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    2013 sat competition, (accessed: January 31, 2014)
  2. 2.
    Ansótegui, C., Bonet, M.L., Giráldez-Cru, J., Levy, J.: The fractal dimension of sat formulas. arXiv preprint arXiv:1308.5046 (2013)Google Scholar
  3. 3.
    Ansótegui, C., Giráldez-Cru, J., Levy, J.: The community structure of sat formulas. In: Cimatti, A., Sebastiani, R. (eds.) SAT 2012. LNCS, vol. 7317, pp. 410–423. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  4. 4.
    Ansótegui, C., Levy, J.: On the modularity of industrial sat instances. In: CCIA, pp. 11–20 (2011)Google Scholar
  5. 5.
    Audemard, G., Simon, L.: Predicting learnt clauses quality in modern SAT solvers. Proceedings of IJCAI, 399–404 (2009)Google Scholar
  6. 6.
    Clauset, A., Newman, M.E.J., Moore, C.: Finding community structure in very large networks. Physical review E 70(6), 66111 (2004)CrossRefGoogle Scholar
  7. 7.
    Eén, N., Sörensson, N.: An extensible sat-solver. In: Giunchiglia, E., Tacchella, A. (eds.) SAT 2003. LNCS, vol. 2919, pp. 502–518. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  8. 8.
    Een, N., Sörensson, N.: Minisat: A sat solver with conflict-clause minimization. In: SAT, vol. 5 (2005)Google Scholar
  9. 9.
    Ian, P.: Gent and Toby Walsh. The sat phase transition. In: ECAI, pp. 105–109. PITMAN (1994)Google Scholar
  10. 10.
    Habet, D., Toumi, D.: Empirical study of the behavior of conflict analysis in cdcl solvers. In: Schulte, C. (ed.) CP 2013. LNCS, vol. 8124, pp. 678–693. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  11. 11.
    Mitchell, D., Selman, B., Levesque, H.: Hard and easy distributions of sat problems. In: AAAI, vol. 92, pp. 459–465. Citeseer (1992)Google Scholar
  12. 12.
    Newman, M.E.J.: Fast algorithm for detecting community structure in networks. Physical review E 69(6), 66133 (2004)CrossRefGoogle Scholar
  13. 13.
    Newsham, Z., Ganesh, V., Fischmeister, S., Audemard, G., Simon, L.: Community Structure of SAT Instances Webpage with Data and Code,
  14. 14.
    Pipatsrisawat, K., Darwiche, A.: A lightweight component caching scheme for satisfiability solvers. In: Marques-Silva, J., Sakallah, K.A. (eds.) SAT 2007. LNCS, vol. 4501, pp. 294–299. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  15. 15.
    Vardi, M.: Phase transition and computation complexity (2012),
  16. 16.
    Xu, L., Hutter, F., Hoos, H.H., Leyton-Brown, K.: Satzilla: Portfolio-based algorithm selection for sat. J. Artif. Intell. Res. (JAIR) 32, 565–606 (2008)Google Scholar
  17. 17.
    Zhang, W., Pan, G., Wu, Z., Li, S.: Online community detection for large complex networks. In: Proceedings of the Twenty-Third International Joint Conference on Artificial Intelligence, pp. 1903–1909. AAAI Press (2013)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Zack Newsham
    • 1
  • Vijay Ganesh
    • 1
  • Sebastian Fischmeister
    • 1
  • Gilles Audemard
    • 2
  • Laurent Simon
    • 3
  1. 1.University of WaterlooWaterlooCanada
  2. 2.Laboratoire Bordelais de Recherche en InformatiqueBordeaux CedexFrance
  3. 3.CRIL - CNRS UMR 8188Université Lille-Nord de FranceLensFrance

Personalised recommendations