K-LSAT is NP-Complete for k≥3

Abstract

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.LCNF_≥k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LSAT_≥k的NP-完全性取决于LCNF_≥k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LCNF_≥3和LCNF_≥4中的不可满足公式,并提出公开问题:对于k≥5,LCNF_≥k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.;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 LCNF_≥kof LCNF,in which formulas have only clauses of length at least k,the NP-completeness of the decision problem LSAT_≥kis closely relevant to whether or not there exists an unsatisfiable formula in LCNF_≥k,i.e.,the NP-eompletness of SAT for LCNF_≥k(k≥3)is the question whether there exists an unsatisfiable formula in LCNF_≥k.S.Porschen et al.have shown that both LCNF_≥3and LCNF_≥4contain 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 LCNF_≥k.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.