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Leveraging belief propagation, backtrack search, and statistics for model counting

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Abstract

We consider the problem of estimating the model count (number of solutions) of Boolean formulas, and present two techniques that compute estimates of these counts, as well as either lower or upper bounds with different trade-offs between efficiency, bound quality, and correctness guarantee. For lower bounds, we use a recent framework for probabilistic correctness guarantees, and exploit message passing techniques for marginal probability estimation, namely, variations of the Belief Propagation (BP) algorithm. Our results suggest that BP provides useful information even on structured, loopy formulas. For upper bounds, we perform multiple runs of the MiniSat SAT solver with a minor modification, and obtain statistical bounds on the model count based on the observation that the distribution of a certain quantity of interest is often very close to the normal distribution. Our experiments demonstrate that our model counters based on these two ideas, BPCount and MiniCount, can provide very good bounds in time significantly less than alternative approaches.

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Authors and Affiliations

  1. Department of Computer Science, Cornell University, Ithaca, NY, 14853-7501, USA

    Lukas Kroc, Ashish Sabharwal & Bart Selman

Authors
  1. Lukas Kroc
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  2. Ashish Sabharwal
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  3. Bart Selman
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Corresponding author

Correspondence to Ashish Sabharwal.

Additional information

A preliminary version of this article appeared at the 5th International Conference on Integration of AI and OR Techniques in Constraint Programming for Combinatorial Optimization Problems (CP-AI-OR), Paris, France, 2008.

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Kroc, L., Sabharwal, A. & Selman, B. Leveraging belief propagation, backtrack search, and statistics for model counting. Ann Oper Res 184, 209–231 (2011). https://doi.org/10.1007/s10479-009-0680-7

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