NP-Completeness of (k-SAT, r-UNk-SAT) and (LSAT>=k, r-UNLSAT>=k)

k-CNF is the class of CNF formulas in which the length of eachclause of every formula is k. The decision problem asks for anassignment of truth values to the variables that satisfies all theclauses of a given CNF formula. k-SAT problem is k-CNF decisionproblem. Cook [9] has shown that k-SAT is NP-complete fork? 3. LCNFis the class of linearformulas and LSATis its decision problem. In [3] wepresent a general method to construct linear minimal unsatisfiable(MU) formulas. NP= PCP(log,1) is calledPCPtheorem, and it is equivalent to that there existssome r> 1 such that (3SAT, r-UN3SAT)(ordenoted by $(1-\frac{1}{r})-GAP3SAT)$ is NP-complete [1,2]. In thispaper,we show that for k? 3, (kSAT,r-UNkSAT) is NP-completre and (LSAT, r-UNLSAT) isNP-completre for some r> 1. Based on the applicationof linear MU formulas, [3] we construct a reduction from (4SAT,r-UN4SAT) to (LSAT≥4,r'-UNLSAT≥ 4), and provedthat (LSAT≥ 4,r- UNLSAT≥ 4) is NP-complete for some r> 1, sothe approximation problem s-Approx-LSAT≥ 4is NP-hard for some s> 1.