Super Solutions of Random Instances of Satisfiability

We study the probabilistic behaviour of super solutions to random instances of the Boolean Satisfiability (SAT) and Constraint Satisfaction Problems (CSPs). Our analysis focuses on a special type of super solutions, the (1,0)-super solutions. For random k-SAT, we establish the exact threshold of the phase transition of the solution probability for the cases of k = 2 and 3, and upper and lower bounds on the threshold of the phase transition for the case of k >= 4. For CSPs, by overcoming difficulties that do not exist in the probabilistic analysis of the standard solution concept, we manage to derive a non-trivial upper bound on the threshold for the probability of having a super solution.