| 王永平,许道云.取定s的严格d-正则随机(3,2s)-SAT问题的可满足临界.软件学报,0,(0):0 |
| 取定s的严格d-正则随机(3,2s)-SAT问题的可满足临界 |
| Satisfiability Threshold of the Strictly d-Regular Random (3,2s)-SAT Problem for Fixed s |
| 投稿时间：2019-03-22 修订日期：2019-11-28 |
| DOI：10.13328/j.cnki.jos.006049 |
| 中文关键词: 3-CNF公式 随机难解实例生成 正则子类 严格d-正则随机(3,2s)-SAT问题 可满足临界 |
| 英文关键词:3-CNF formula generating random hard instances subclass with regular structure strictly d-regular random (3,2s)-SAT problem satisfiability threshold |
| 基金项目:国家自然科学基金（61762019，61862051） |
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| 中文摘要: |
| 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. |
| 英文摘要: |
| 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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