Abstract
Recently, there has been an interest in studying non-uniform random k-satisfiability (k-SAT) models in order to address the non-uniformity of formulas arising from real-world applications. While uniform random k-SAT has been extensively studied from both a theoretical and experimental perspective, understanding the algorithmic complexity of heterogeneous distributions is still an open challenge. When a sufficiently dense formula is guaranteed to be satisfiable by conditioning or a planted assignment, it is well-known that uniform random k-SAT is easy on average. We generalize this result to the broad class of non-uniform random k-SAT models that are characterized only by an ensemble of distributions over variables with a mild balancing condition. This balancing condition rules out extremely skewed distributions in which nearly half the variables occur less frequently than a small constant fraction of the most frequent variables, but generalizes recently studied non-uniform k-SAT distributions such as power-law and geometric formulas. We show that for all formulas generated from this model of at least logarithmic densities, a simple greedy algorithm can find a solution with high probability.
As a side result we show that the total variation distance between planted and filtered (conditioned on satisfiability) models is o(1) once the planted model produces formulas with a unique solution with probability \(1-o(1)\). This holds for all random k-SAT models where the signs of variables are drawn uniformly and independently at random.
Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – 416061626.
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Notes
- 1.
We say that an event \(\mathcal {E}\) holds asymptotically almost surely (a. a. s.) if, over a sequence of sets, \(\Pr \left( \mathcal {E}\right) =1\). In the context of this paper, this means \(\Pr \left( \mathcal {E}\right) =1-o(1)\).
- 2.
We refer to the geometric degree-distribution model introduced by Ansótegui et al. [6].
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Friedrich, T., Neumann, F., Rothenberger, R., Sutton, A.M. (2021). Solving Non-uniform Planted and Filtered Random SAT Formulas Greedily. In: Li, CM., Manyà, F. (eds) Theory and Applications of Satisfiability Testing – SAT 2021. SAT 2021. Lecture Notes in Computer Science(), vol 12831. Springer, Cham. https://doi.org/10.1007/978-3-030-80223-3_13
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