On SAT Distributions with Planted Assignments

While it is known how to generate satisfiable instances by reducing certain computational problems to SAT, it is not known how a similar generator can be developed directly for k-SAT. In this work we almost answer this question affirmatively by improving upon previous results in many ways. First, we give a generator for instances of MAX k-SAT, the version of k-SAT where one wants to maximize the number of satisfied clauses. Second, we provide a useful characterization of the optimal solution. In our model not only we know how the optimal solution looks like but we also prove it is unique. Finally, we show that our generator has certain useful computational properties among which is the ability to control the hardness of the generated instances, the appearance of an easy-hard-easy pattern in the search complexity for good assignments and a new type of phase transition which is related to the uniqueness of the optimal solution.