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Journal of Software:2016.27(12):2985-2993

随机正则(k,r)-SAT问题的可满足临界
周锦程,许道云,卢友军
(贵州大学 计算机科学与技术学院, 贵州 贵阳 550025;黔南民族师范学院 数学与统计学院, 贵州 都匀 558000)
Satisfiability Threshold of the Regular Random (k,r)-SAT Problem
ZHOU Jin-Cheng,XU Dao-Yun,LU You-Jun
(College of Computer Science and Technology, Guizhou University, Guiyang 550025, China;School of Mathematics and Statistics, Qiannan Normal University for Nationalities, Duyun 558000, China)
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Received:July 05, 2016    
> 中文摘要: 研究k-SAT问题实例中每个变元恰好出现r=2s次,且每个变元对应的正、负文字都出现s次的严格随机正则(k,r)-SAT问题.通过构造一个特殊的独立随机实验,结合一阶矩方法,给出了严格随机正则(k,r)-SAT问题可满足临界值的上界.由于严格正则情形与正则情形的可满足临界值近似相等,因此得到了随机正则(k,r)-SAT问题可满足临界值的新上界.该上界不仅小于当前已有的随机正则(k,r)-SAT问题的可满足临界值上界,而且还小于一般的随机k-SAT问题的可满足临界值.因此,这也从理论上解释了在相变点处的随机正则(k,r)-SAT问题实例通常比在相应相变点处同规模的随机k-SAT问题实例更难满足的原因.最后,数值分析结果验证了所给上界的正确性.
Abstract:This article studies the strictly regular (k,r)-SAT problem by restricting the k-SAT problem instances, where each variables occurs precisely r=2s times and each of the positive and negative literals occurs precisely s times. By constructing a special independent random experiment, the study derives an upper bound on the satisfiability threshold of the strictly regular random (k,r)-SAT problem via the first moment method. Based on the fact that the satisfiability threshold of the strictly regular and the regular random (k,r)-SAT problems are approximately equal, a new upper bound on the threshold of the regular random (k,r)-SAT problem is obtained. This new upper bound is not only below the current best known upper bounds on the satisfiability threshold of the regular random (k,r)-SAT problem, but also below the satisfiability threshold of the uniform random k-SAT problem. Thus, it is theoretically explained that in general the regular random (k,r)-SAT instances are harder to satisfy at their phase transition points than the uniform random k-SAT problem at the corresponding phase transition points with same scales. Finally, numerical results verify the validity of our new upper bound.
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基金项目:国家自然科学基金(61262006,61463044,61462001);贵州省科技厅联合基金(LKQS201313) 国家自然科学基金(61262006,61463044,61462001);贵州省科技厅联合基金(LKQS201313)
Foundation items:National Natural Science Foundation of China (61262006, 61463044, 61462001); Science and Technology Foundation of Guizhou Province of China (LKQS201313)
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周锦程,许道云,卢友军.随机正则(k,r)-SAT问题的可满足临界.软件学报,2016,27(12):2985-2993

ZHOU Jin-Cheng,XU Dao-Yun,LU You-Jun.Satisfiability Threshold of the Regular Random (k,r)-SAT Problem.Journal of Software,2016,27(12):2985-2993