(2+f(n))-SAT and its properties Original Research Article

Consider a formula that contains n variables with the form Φ=Φ2∧Φ3, where Φ2 is an instance of 2-SAT containing m2 2-clauses and Φ3 is an instance of 3-SAT containing m3 3-clauses. Φ is an instance of (2+f(n))-SAT if m3/(m2+m3)⩽f(n). We prove that (2+f(n))-SAT is in P if f(n)=O(log n/n2), and in NPC if f(n)=1/n2−ε(∀ε: 0<ε<2). Most interestingly, we give a candidate, (2+(log n)k/n2)-SAT (k⩾2), for natural problems in NP−NPC−P (denoted as NPI) with respect to this (2+f(n))-SAT model. We prove that the restricted version of it is not in NPC under P ≠ NP. Actually, it is indeed in NPI under some stronger but plausible assumption, specifically, the exponential-time hypothesis.