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Chapter 0: Q24P (page 1)

Let CNFk= is a satisfiable cnf-formula where each variable appears in at most k places}.

a. Show thatCNF2?P .

b. Show thatCNF3鈥�isNP-complete.

Short Answer

Expert verified

It is clear that蠒is satisfiable if and only if ry(鉄�蠒鉄�)is satisfiable. The ry is a reduced polynomial time in terms of the number of variable in蠒 fromCW_3isNP- complete.

Step by step solution

01

Step 1:Chemical molecule of the equation

Now have to show that CMK2鈭�P-.

LetTybe the polynomial tine decider forCH2-

Tpcan be described as {TwS:

T3=on input鉄�蠒鉄�=

02

Converting variable into polynomial

According to CNF rules,

Consider the first choose of蠒蠒. If it is of the from x, and there isx鈥�in鈥�蠒 , reject

Solve CNF where c occurs in every choose, where negation of c does not appear-

Every time T1processes each variable and reaches either accept or reject Because of this the number of蠒 might decrease by 1 or Hence running time ofTp becomes polynomial tine in terms of the number of variables.

So,CH2鈭�P.

Thus,GW_3isNP- complete

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