单调3- Sat -（2，2）是NP完全的
Monotone 3-Sat-(2,2) is NP-complete
摘要：
我们证明了单调3-Sat是NP-完全的，如果（i）每个子句恰好包含三个不同的变量，（ii）每个子句是唯一的，即，不存在相同子句的重复，以及（iii）在子句中，每个变量恰好出现两次未求反和恰好出现两次求反。Darmann和Döcker [6]最近证明了单调3-Sat的这个变体是平凡的或NP完全的。在本文的第一部分中，我们构造了一个不可满足的实例，它回答了他们的一个开放问题（挑战1），并将问题放在后一类中。然后，我们调整构造中使用的小工具以（1）以更直接的方式绘制建立NP完全性的两个约简，以及（2），以表明$\forall\exists$ 3-SAT对于具有以下性质的量化布尔公式仍然是$Π_2^P$ -完全的：（a）每个子句是单调的（即，没有子句包含非求反和求反变量）并且包含正好三个不同的变量，（b）每个通用变量正好出现一次非求反和正好出现一次求反，（c）每个存在变量正好出现两次非求反和正好出现两次求反，以及（d）通用变量和存在变量的数目相等。此外，我们还证明了当（b）被（b '）替换时，每个泛变量恰好出现两次非否定和两次否定，并且（a），（c）和（d）不变，这个变体也是$Π_2^P$ -完备的。因此，我们改进了Döcker等人最近的两个结果。[8]在非单调设置中建立了这些变体的$Π_2^P$ -完备性。我们还讨论了单调3-Sat-（2，2）的一个特殊情况，它对应于不全相等Sat的一个变体，并且我们证明了所有这样的实例都是可满足的。
英文摘要：
We show that Monotone 3-Sat remains NP-complete if (i) each clause contains exactly three distinct variables, (ii) each clause is unique, i.e., there are no duplicates of the same clause, and (iii), amongst the clauses, each variable appears unnegated exactly twice and negated exactly twice. Darmann and Döcker [6] recently showed that this variant of Monotone 3-Sat is either trivial or NP-complete. In the first part of the paper, we construct an unsatisfiable instance which answers one of their open questions (Challenge 1) and places the problem in the latter category. Then, we adapt gadgets used in the construction to (1) sketch two reductions that establish NP-completeness in a more direct way, and (2), to show that $\forall\exists$ 3-SAT remains $Π_2^P$-complete for quantified Boolean formulas with the following properties: (a) each clause is monotone (i.e., no clause contains an unnegated and a negated variable) and contains exactly three distinct variables, (b) each universal variable appears exactly once unnegated and exactly once negated, (c) each existential variable appears exactly twice unnegated and exactly twice negated, and (d) the number of universal and existential variables is equal. Furthermore, we show that the variant where (b) is replaced with (b') each universal variable appears exactly twice unnegated and exactly twice negated, and where (a), (c) and (d) are unchanged, is $Π_2^P$-complete as well. Thereby, we improve upon two recent results by Döcker et al. [8] that establish $Π_2^P$-completeness of these variants in the non-monotone setting. We also discuss a special case of Monotone 3-Sat-(2,2) that corresponds to a variant of Not-All-Equal Sat, and we show that all such instances are satisfiable.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
