Learning from Survey Propagation: a Neural Network for MAX-E-$3$-SAT

Many natural optimization problems are NP-hard, which implies that they are probably hard to solve exactly in the worst-case. However, in practice, it suffices to get reasonably good solutions for all (or even most) instances. This paper presents a new algorithm for computing approximate solution in ${\Theta(N})$ for the MAX-E-$3$-SAT problem by using deep learning methodology. This methodology allows us to create a learning algorithm able to fix Boolean variables by using local information obtained by the Survey Propagation algorithm. By performing an accurate analysis, on random CNF instances of the MAX-E-$3$-SAT with several Boolean variables, we show that this new algorithm, avoiding any decimation strategy, can build assignments better than a random one, even if the convergence of the messages is not found. Although this algorithm is not competitive with state-of-the-art MAX-SAT solvers, it can solve substantially larger and more difficult problems than it ever saw during training.