Super solutions of random (3 + p)-SAT

In this paper, we propose a model named random (3+p)-SAT, where a random formula ϕ=ϕ(n,r,p) contains n variables and rn clauses, and (1−p)rn of the clauses are 3-clauses and prn of the clauses are 4-clauses. This paper studies the (1,0)-satisfiability of random (3+p)-SAT and obtains rigorous results that the exact (1,0)-satisfiability threshold is rp⁎=1/3(1−p) if p≤3/7. For p≥3/7, we give lower and upper bounds of the (1,0)-satisfiability threshold, where the lower bound is obtained by using the Unit-Clause algorithm, and the upper bound is obtained by using a novel way to count precisely the subset of all (1,0)-satisfying assignments that satisfy a “locally maximum” condition.