Satisfiability Threshold of the Strictly d-Regular Random (3,2s)-SAT Problem for Fixed s

DOI：10.13328/j.cnki.jos.006049

 作者 单位 E-mail 王永平 贵州大学 计算机科学与技术学院, 贵州 贵阳 550025贵州财经大学 数统学院, 贵州 贵阳 550025 许道云 贵州大学 计算机科学与技术学院, 贵州 贵阳 550025 dyxu@gzu.edu.cn

3-CNF公式的随机难解实例生成对于揭示3-SAT问题的难解实质和设计满足性测试的有效算法有着重要意义.对于整数k>2和s>2如果在一个k-CNF公式中每个变量正负出现次数均为s，则称该公式是严格正则（k，2s）-CNF公式.受严格正则（k，2s）-CNF公式的结构特征启发，提出每个变量正负出现次数之差的绝对值均为d的严格d-正则（k，2s）-CNF公式，并使用新提出的SDRRK2S模型生成严格d-正则随机（k，2s）-CNF公式.取定整数5 < s < 11模拟实验显示，严格d-正则随机（3，2s）-SAT问题存在SAT-UNSAT相变现象和HARD-EASY相变现象.因此，立足于3-CNF公式的随机难解实例生成，研究了严格d-正则随机（3，2s）-SAT问题在s取定时的可满足临界.通过构造一个特殊随机试验和使用一阶矩方法得到了严格d-正则随机（3，2s）-SAT问题在s取定时可满足临界值的一个下界.模拟实验结果验证了理论证明所得下界的正确性.

Generating random hard instances of the 3-CNF formula is an important factor in revealing the intractability of the 3-SAT problem and designing effective algorithms for satisfiability testing. Let k > 2 and s > 0 be integers, a k-CNF formula is a strictly regular (k, 2s)-CNF one if the positive and negative occurrence number of every variable in the formula are s. On the basis of the strictly regular (k, 2s)-CNF formula, we propose the strictly d-regular (k, 2s)-CNF formula in which the absolute value of the difference between positive and negative occurrence number of every variable is d. We construct a novel model to generate the strictly d-regular random (k, 2s)-CNF formula. Our simulated experiments show that the strictly d-regular random (3, 2s)-SAT problem has an SAT-UNSAT phase transition and an HARD-EASY phase transition when the parameter 5 < s < 11 is fixed, and that the latter is related to the former. Hence, we study the satisfiability threshold of the strictly d-regular random (3, 2s)-SAT problem when the parameter s is fixed. We obtain a lower bound of the satisfiability threshold by constructing a random experiment and using the first moment method. The subsequent simulated experiments verify well the lower bound proved by us.
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