Model counting with error-correcting codes

Abstract

The idea of counting the number of satisfying truth assignments (models) of a formula by adding random parity constraints can be traced back to the seminal work of Valiant and Vazirani showing that NP is as easy as detecting unique solutions. While theoretically sound, the random parity constraints used in that construction suffer from the following drawback: each constraint, on average, involves half of all variables. As a result, the branching factor associated with searching for models that also satisfy the parity constraints quickly gets out of hand. In this work we prove that one can work with much shorter parity constraints and still get rigorous mathematical guarantees, especially when the number of models is large so that many constraints need to be added. Our work is motivated by the realization that the essential feature for a system of parity constraints to be useful in probabilistic model counting is that its set of solutions resembles an error-correcting code.

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Notes

  1. 1.

    This can be done by selecting a uniformly random permutation of size [n] and using it to map each of the n non-zeros to equations; when , ∈ O(1), the variables in each equation will be distinct with probability Ω(1), so that a handful of trials suffice to generate a matrix as desired.

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Acknowledgements

We are grateful to Kuldeep Meel and Moshe Vardi for sharing their code and formulas and for several valuable conversations. We also thank Zayd Hammoudeh, Ben Sherman, and Kostas Zampetakis for several comments on earlier versions.

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Correspondence to Panos Theodoropoulos.

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Research supported by European Research Council Starting Grant 210743, NSF grant CCF-1514128, NSF CCF-1733884, an Adobe research grant, and the Greek State Scholarships Foundation (IKY).

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Achlioptas, D., Theodoropoulos, P. Model counting with error-correcting codes. Constraints 24, 162–182 (2019). https://doi.org/10.1007/s10601-018-9296-3

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Keywords

  • Model counting
  • Satisfiability
  • Randomized algorithms
  • Systems of parity equations
  • Coding theory
  • Low density parity check codes