In the exact 4sat problem, the input is a set of clauses, each of which is a disjunction of exactly four literals, and such that each variable occurs at most once in each clause. the goal is to nd a satisfying assignment, if one exists. prove that exact 4sat is np-complete.

To prove that 4-SAT is NP-complete, we need to prove that 4-SAT is in NP and NP-hard both.

Case 1:

We can write a non-deterministic algorithm that will take an instance of 4-SAT and a truth assignment. Then it would verify if the instance evaluates to true.

This algorithm will take polynomial time. So 4-SAT is in NP.

Case 2:

Now let us reduce 3-SAT to 4-SAT, to prove 4-SAT in NP-hard.

Let say we have a 3-SAT instance C, where each clause is consisting of three literals.

Now to find the equivalent 4-SAT C’, replace each clause (p ∨ q ∨ r ) as (p ∨ q ∨ r ∨ s) ∧ (p ∨ q ∨ r ∨  ¬s).

•If the 3-SAT C is satisfiable by a truth assignment, then also C’ is satisfiable. As if we have the truth assignment for (p ∨ q ∨ r ), then we can find the same for (p ∨ q ∨ r ∨ s) ∧ (p ∨ q ∨ r ∨  ¬s).

•And if there is some truth assignment for C’, then that C will be satisfiable under the same assignment. Because if (p ∨ q ∨ r ∨ s) ∧ (p ∨ q ∨ r ∨  ¬s) is true then (p ∨ q ∨ r ) can be true.

Hence 4-SAT is in NP-hard.  

Due to both cases, 4-SAT is NP-complete.

i think so.

by: nicholas

the benefits of raising bc’s minimum wage to $15 through a

series of pre-announced staged increases far outweigh the likely

costs. claims that such an increase will lead to massive job losses

in low-wage sectors of the economy are not credible.

• an increase to $15 would result in some job losses. however,

the likely impact is less than a 1 per cent reduction in the overall

provincial employment rate, and a 7.6 per cent reduction in the

employment rate for those currently making between $10.25

and $15 per hour.

• the large job loss impacts predicted by some opponents of

minimum wages misrepresent the existing economic research. in

reading and using that research, it is important to recognize that

estimates showing job loss effects of minimum wage increases

apply only to teenagers. estimated effects for young adult and

adult workers range from insignificant to non-existent.

• focusing on teenage workers plays up an inaccurate stereotype

about who earns very low wages in bc and leads to exaggerated

claims about job losses. only 26 per cent of those currently making

between $10.25 and $12 are teenagers, and only 5 per cent

of those making between $12 and $15 are teenagers.