(2 + f(n))-SAT and Its Properties

Abstract

Consider a formula which contains n variables and m clauses with the form Φ = Φ ₂ Λ Φ ₃ , where Φ ₂ is an instance of 2-SAT which contains m ₂ 2-clauses and Φ ₃ is an instance of 3-SAT which contains m3 3-clauses. Фis an instance of (2 + f(n))-SAT if m ₃/m ₂+m ₃≤ f(n). We prove that (2 + f(n))-SAT is in $\mathcal{P}$ if f(n) = O (log n /n ²), and in $\mathcal{N}\mathcal{P}\mathcal{C}$ if f(n) = 1/n ^2-ɛ(∨ɛ: 0 < ε < 2). Most interestingly, we give a candidate ( 2 + (log n)^k/n ²)-SAT (k ≥ 2), for natural problems in $\mathcal{N}\mathcal{P}--\mathcal{N}\mathcal{P}\mathcal{C}--\mathcal{P}$ (denoted as $\mathcal{N}\mathcal{P}\mathcal{I}$ ) with respect to this (2 + f(n))-SAT model. We prove that the restricted version of it is not in $\mathcal{N}\mathcal{P}\mathcal{C}$ under the assumption $\mathcal{P}$ ≠$\mathcal{N}\mathcal{P}$ . Actually it is indeed in $\mathcal{N}\mathcal{P}\mathcal{I}$ under some stronger but plausible assumption, specifically, the Exponential-Time Hypothesis (ETH) which was introduced by Impagliazzo and Paturi.

This research is supported by a research grant of City University of Hong Kong 7001023.