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.
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.
作者
周锦程
许道云
卢友军
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)
出处
《软件学报》
EI
CSCD
北大核心
2016年第12期2985-2993,共9页
Journal of Software
基金
国家自然科学基金(61262006
61463044
61462001)
贵州省科技厅联合基金(LKQS201313)~~
关键词
随机正则(k
r)-SAT问题
可满足临界值
相变现象
计算复杂性
regular random（k
r）-SAT problem
satisfiability threshold
phase transition phenomena
computational complexity
