# GATE2014-3-38

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Consider the decision problem $2CNFSAT$ defined as follows:

$$\left\{ \phi \mid \phi \text{ is a satisfiable propositional formula in CNF with at most two literals per clause}\right\}$$

For example, $\phi = (x1 \vee x2) \wedge (x1 \vee \bar{x3}) \wedge (x2 \vee x4)$ is a Boolean formula and it is in $2CNFSAT$.

The decision problem $2CNFSAT$ is

1. NP-complete
2. solvable in polynomial time by reduction to directed graph reachability.
3. solvable in constant time since any input instance is satisfiable.
4. NP-hard but not NP-complete.

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2CNF is P.

SO A) & D) are ruled out.

C) solvable in constant time since any input instance is satisfiable. => This is false. Input instanece where this is false x1=F, x2=F

Then it becomes false.

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C) is not false because of your given reason. Satisfiability means that you can have an assignment of values which make the expression true which you can clearly do for the given expression. It's false since you could have an expression like (A AND ~A) which always evaluates to false and is thus not satisfiable.
this problem is directly reduced to strongly connected component and hence B.

2CNF-SAT can be reduced to strongly connected components problem. And strongly connected component has a polynomial time solution. Therefore 2CNF-SAT is polynomial time solvable. See https://en.wikipedia.org/wiki/2-satisfiability#Strongly_connected_componentsfor details.

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