一类可分离SAT问题的O(1.890n)精确算法

布尔可满足性问题（SAT）是指对于给定的布尔公式，是否存在一个可满足的真值指派.这是第1个被证明的NP完全问题，一般认为不存在多项式时间算法，除非P=NP.学者们大都研究了子句长度不超过k的SAT问题（k-SAT），从全局搜索到局部搜索，给出了大量的相对有效算法，包括随机算法和确定算法.目前，最好算法的时间复杂度不超过O（（2-2/k）ⁿ），当k=3时，最好算法时间复杂度为O（1.308ⁿ）.而对于更一般的与子句长度k无关的SAT问题，很少有文献涉及.引入了一类可分离SAT问题，即3-正则可分离可满足性问题（3-RSSAT），证明了3-RSSAT是NP完全问题，给出了一般SAT问题3-正则可分离性的O（1.890ⁿ）判定算法.然后，利用矩阵相乘算法的研究成果，给出了3-RSSAT问题的O（1.890ⁿ）精确算法，该算法与子句长度无关.

O(1.890n) Exact Algorithm for a Class of Separable SAT Problems

The Boolean satisfiability problem (SAT) refers to whether there is a truth assignment that satisfies a given Boolean formula, which is the first confirmed NP complete problem that generally does not exist a polynomial time algorithm unless P=NP. However many practical applications of such problems often take place and are in need of an effective algorithm to reduce their time complexity. At present, many scholars have studied the problem of SAT with clause length not exceeding k (k-SAT). From global search to local search, a large number of effective algorithms, including random algorithm and determination algorithm are developed, and the best result, including probabilistic algorithm and deterministic algorithm for solving k-SAT problems, is that the time complexity is less than O((2-2/k)ⁿ), and when k=3 the time complexity of the best algorithm is O(1.308ⁿ). However, there is little literature about SAT problems that are more general than clause length k. This paper discusses a class of separable satisfiability problems (SSAT), in particular, the problem of 3-regular separable satisfiability (3-RSSAT) where the formula can be separated into several subformulas according to certain rules. The paper proves that 3-RSSAT problem is NP complete problem because any SAT problem can be polynomially reduced to it. To determine 3-regular separability of the general SAT problem, an algorithm is given with time complexity is no more than O(1.890ⁿ). Then by using the result in the matrix multiplication algorithm optimal research field, an O(1.890ⁿ) exact algorithm is constructed for solving the 3-RSSAT problem, which is the WELL algorithm independent of clause length.