[Solved]: SAT not reducible to 2SAT

Problem Detail: 

Why is the reduction $\textbf{SAT} \leq_P \textbf{3SAT}$ possible, but $\textbf{SAT} \leq_P \textbf{2SAT}$ not possible, given, that $\textbf{SAT}$ is $\textbf{NP}$-complete, $\textbf{2SAT} \in \textbf{NP}$ and $\textbf{3SAT} \in \textbf{NP}$?

Asked By : Anderson

Answered By : mdxn

I believe the asker is wanting to know specifically why the standard approach to reducing $\text{SAT}$ to $\text{3SAT}$ does not continue to extend to the $\text{2SAT}$ case. Here is a walkthrough as to why.

Conversion of a SAT Instance to a 3-SAT Instance:

For convenience, let us assume everything is in $\text{CNF}$, or conjunctive normal form. The $\text{SAT} \le_p \text{3SAT}$ proof works by introducing dummy variables to spread the actual variables over multiple clauses.

Illustration: Consider a single clause in boolean formula $(x_1 \lor x_2 \lor x_3 \lor x_4)$. The $\text{3SAT}$ reduction works by introducing a dummy variable to represent a disjunction of two literals in hopes that it reduces the clause's length by one. For example, $y = x_3\lor x_4$. The clause can be rewritten as:

$(x_1 \lor x_2 \lor y) \land (\bar y \lor x_3 \lor x_4)$

Notice that these clauses are now in a $\text{CNF}$ for $\text{3SAT}$. If this is repeated multiple times for each clause, the final $\text{CNF}$ formula will be at most polynomial in length of the original. This is why a polynomial time reduction would suffice.

Attempting to Apply the Method to Convert to 2-SAT:

If we tried to use the trick above with a clause of length 3, $(a\lor b\lor c)$, then we would not get the desired improvement. Let our dummy variable here be $y=b \lor c$.

The result: $(a \lor y) \land (\bar y \lor b \lor c)$

As you can see, it still has a clause of size 3. If we used the method again on the second clause, we would end up with the same problem while also adding an additional clause of length 2.

It could be said that the simplest reason for this is because $\land$ and $\lor$ are binary operators. You still need to apply these operators on pairs of variables, then include an additional dummy variable (resulting in a clause of length 3).

There have been attempts at getting around this that are successful, but the resulting sequence of clauses become exponential in length. Handling such a formula is outside the scope of $\text{P}$, which is why such a reduction would not be a polynomial time reduction.

Best Answer from StackOverflow

Question Source : http://cs.stackexchange.com/questions/14596