Abstract
The ability to sample solutions of a constrained combinatorial space has important applications in areas such as probabilistic reasoning and hardware/software verification. A highly desirable property of such samples is that they should be drawn uniformly at random, or at least nearly so. For combinatorial spaces expressed as sat models, approaches based on universal hashing provide probabilistic guarantees about sampling uniformity. In this short paper, we apply that same approach to cp models, for which hashing functions take the form of linear constraints in modular arithmetic. We design an algorithm to generate an appropriate combination of linear modular constraints given a desired sample size. We evaluate empirically the sampling uniformity and runtime efficiency of our approach, showing it to be near-uniform at a fraction of the time needed to draw from the complete set of solutions.
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Notes
- 1.
i.e. each congruent to one of \(\{0,1,\ldots ,c[i]\}\). We need to add \(\boldsymbol{b}\) in the inequalities because otherwise the probability that e.g. the null solution \(\boldsymbol{x=0}\) satisfies the inequality would be equal to 1.
- 2.
- 3.
The sampling method is implemented in the MiniCPBP solver https://github.com/PesantGilles/MiniCPBP and our examples on how to use the method are available https://github.com/363734/UniformSampling.
- 4.
The statistical library used is “Apache Commons Mathematics Library” (https://commons.apache.org/proper/commons-math/).
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Acknowledgements
We thank the anonymous reviewers for their constructive criticism which helped us improve the original version of the paper. Financial support for this research was provided in part by NSERC Discovery Grants 218028/2017 and 05953/2016.
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Pesant, G., Quimper, CG., Verhaeghe, H. (2022). Practically Uniform Solution Sampling in Constraint Programming. In: Schaus, P. (eds) Integration of Constraint Programming, Artificial Intelligence, and Operations Research. CPAIOR 2022. Lecture Notes in Computer Science, vol 13292. Springer, Cham. https://doi.org/10.1007/978-3-031-08011-1_22
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