非全等3-Sat和3-Sat的简化NP完全变式
On simplified NP-complete variants of Not-All-Equal 3-Sat and 3-Sat
摘要：
我们考虑简化的，单调版本的不全相等3-Sat和3-Sat，著名的可满足性问题的变种，其中每个子句是由三个不同的文字。我们证明了，即使（1）每个变量恰好出现四次，（2）公式中没有否定，（3）公式是线性的，即，每对不同的子句最多共享一个变量。关于3-Sat，我们证明了几个硬度结果的单调公式就施加各种限制的变量外观。单调3-Sat是3-Sat对单调公式的限制，即每个子句分别仅包含非否定变量或仅包含否定变量的公式。特别是，我们表明，对于任何$k\geq 5$，单调3-Sat是NP-完全的，即使每个变量恰好出现$k$次未否定和恰好一次否定。此外，我们证明了单调3-Sat是NP-完全的，即使每个变量分别出现三次非否定和三次否定。事实上，我们提供了Monotone 3-Sat的完整分析，每个变量恰好出现六次。此外，我们证明了这个问题仍然是NP-完全的限制时，每个变量出现的情况下，要么正好一次非否定和三次否定或其他方式。因此，我们改进了Darmann等人的结果。[DDD 18]显示每个变量四次出现的NP完全性。我们更强的结果还意味着3-Sat仍然是NP完全的，即使每个变量恰好出现三次未否定和一次否定，从而补充了Berman等人的结果。[BKS 03]。
英文摘要：
We consider simplified, monotone versions of Not-All-Equal 3-Sat and 3-Sat, variants of the famous Satisfiability Problem where each clause is made up of exactly three distinct literals. We show that Not-All-Equal 3-Sat remains NP-complete even if (1) each variable appears exactly four times, (2) there are no negations in the formula, and (3) the formula is linear, i.e., each pair of distinct clauses shares at most one variable. Concerning 3-Sat we prove several hardness results for monotone formulas with respect to a variety of restrictions imposed on the variable appearances. Monotone 3-Sat is the restriction of 3-Sat to monotone formulas, i.e. to formulas in which each clause contains only unnegated variables or only negated variables, respectively. In particular, we show that, for any $k\geq 5$, Monotone 3-Sat is NP-complete even if each variable appears exactly $k$ times unnegated and exactly once negated. In addition, we show that Monotone 3-Sat is NP-complete even if each variable appears exactly three times unnegated and three times negated, respectively. In fact, we provide a complete analysis of Monotone 3-Sat with exactly six appearances per variable. Further, we prove that the problem remains NP-complete when restricted to instances in which each variable appears either exactly once unnegated and three times negated or the other way around. Thereby, we improve on a result by Darmann et al. [DDD18] showing NP-completeness for four appearances per variable. Our stronger result also implies that 3-Sat remains NP-complete even if each variable appears exactly three times unnegated and once negated, therewith complementing a result by Berman et al. [BKS03].
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