k-LSAT (k≥3)是NP-完全的
(贵州大学 计算机科学系,贵州 贵阳 550025)
k-LSAT is NP-Complete for k≥3
XU Dao-Yun,DENG Tian-Yan,ZHANG Qing-Shun
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Received:December 13, 2006    Revised:January 24, 2007
> 中文摘要: 合取范式(conjunctive normal form,简称CNF)公式F是线性公式,如果F中任意两个不同子句至多有一个公共变元.如果F中的任意两个不同子句恰好含有一个公共变元,则称F是严格线性的.所有的严格线性公式均是可满足的,而对于线性公式类LCNF,对应的判定问题LSAT仍然是NP-完全的.LCNFk是子句长度大于或等于k的CNF公式子类,判定问题LSA(≥k)的NP-完全性与LCNF(≥k)中是否含有不可满足公式密切相关.即LSATk的NP-完全性取决于LCNFk是否含有不可满足公式.S.Porschen等人用超图和拉丁方的方法构造了LCNF3和LCNF4中的不可满足公式,并提出公开问题:对于k≥5,LCNFk是否含有不可满足公式?将极小不可满足公式应用于公式的归约,引入了一个简单的一般构造方法.证明了对于k≥3,k-LCNF含有不可满足公式,从而证明了一个更强的结果:对于k≥3,k-LSAT是NP-完全的.
Abstract:A CNF formula F is linear if any distinct clauses in F contain at most one common variable.A CNF formula F is exact linear if any distinet clauses in F contain exactly one conlrnon vailable.All exact linear formulas are satisfiable[1],and for the class LCNF of linear formulas,the decision problem LSAT remains NP-complete.For the subclasses LCNFkof LCNF,in which formulas have only clauses of length at least k,the NP-completeness of the decision problem LSATkis closely relevant to whether or not there exists an unsatisfiable formula in LCNFk,i.e.,the NP-eompletness of SAT for LCNFk(k≥3)is the question whether there exists an unsatisfiable formula in LCNFk.S.Porschen et al.have shown that both LCNF3and LCNF4contain unsatisfiable formulas by the constructions of hypergraphs and latin squares.It leaves the open question whether for each k≥5 there is an unsatisfiable formula in LCNFk.This paper presents a simple and general method to construct unsatisfiable formulas in k-LCNF for each k≥3 by the application of minimal unsatisfiable formulas to reductions for formulas.It is shown that for each k≥3 there exists a minimal unsatisfiable formula in k-LCNF.Therefore,the stronger result is shown that k-LSAT is NP-complete for k≥3.
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基金项目:Supported by the National Natural Science Foundation of China under Grant Nos.10410638,60310213(国家自然科学基金) Supported by the National Natural Science Foundation of China under Grant Nos.10410638,60310213(国家自然科学基金)
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许道云,邓天炎,张庆顺.k-LSAT (k≥3)是NP-完全的.软件学报,2008,19(3):511-521

XU Dao-Yun,DENG Tian-Yan,ZHANG Qing-Shun.k-LSAT is NP-Complete for k≥3.Journal of Software,2008,19(3):511-521