算法设计期末作业02-8.8

题目：

In theEXACT4SATproblem, the input is a set of clauses, each of which is a disjunction of exactly
four literals, and such that each variable occurs at most once in each clause. The goal is to find
a satisfying assignment, if one exists. Prove that EXACT4SATis NP-complete.

解答过程：

首先，因为3SAT是NP-complete问题，下面只需要把3SAT问题在多项式时间内规约到4SAT问题上，即可证明4SAT问题是NP-complete问题。因为（a1∪a2∪a3）==（a1∪a2∪a3∪a4）∩（a1∪a2∪a3∪~a4），当a1∪a2∪a3为真时，（a1∪a2∪a3∪a4）∩（a1∪a2∪a3∪~a4）必为真，且a4和~a4必有一个是假的。假的语句对应的前面的a1∪a2∪a3必为真。同理，对于（a1∪a2），可以转为（a1∪a2∪a3∪a4）∩（a1∪a2∪a3∪~a4）∩（a1∪a2∪~a3∪a4）∩（a1∪a2∪~a3∪~a4）。当a1∪a2为真时，后面部分a3和a4不管怎么搭配，必为3个true，1个false,对于false的那个，其前面a1∪a2必为true。对于a1同理。
故综上所述，EXACT4SATis NP-complete