Learning-Augmented Algorithms for Boolean Satisfiability

Learning-augmented algorithms are a prominent recent development in beyond worst-case analysis. In this framework, a problem instance is provided with a prediction (``advice'') from a machine-learning oracle, which provides partial information about an optimal solution, and the goal is to design algorithms that leverage this advice to improve worst-case performance. We study the classic Boolean satisfiability (SAT) decision and optimization problems within this framework using two forms of advice. ``Subset advice" provides a random $\epsilon$ fraction of the variables from an optimal assignment, whereas ``label advice" provides noisy predictions for all variables in an optimal assignment. For the decision problem $k$-SAT, by using the subset advice we accelerate the exponential running time of the PPSZ family of algorithms due to Paturi, Pudlak, Saks and Zane, which currently represent the state of the art in the worst case. We accelerate the running time by a multiplicative factor of $2^{-c}$ in the base of the exponent, where $c$ is a function of $\epsilon$ and $k$. For the optimization problem, we show how to incorporate subset advice in a black-box fashion with any $\alpha$-approximation algorithm, improving the approximation ratio to $\alpha + (1 - \alpha)\epsilon$. Specifically, we achieve approximations of $0.94 + \Omega(\epsilon)$ for MAX-$2$-SAT, $7/8 + \Omega(\epsilon)$ for MAX-$3$-SAT, and $0.79 + \Omega(\epsilon)$ for MAX-SAT. Moreover, for label advice, we obtain near-optimal approximation for instances with large average degree, thereby generalizing recent results on MAX-CUT and MAX-$2$-LIN.