# P-NP Theory

 Question 1
The problem 3-SAT and 2-SAT are
 A both in P B both NP complete C NP -complete and in P respectively D undecidable and NP complete respectively
ISRO CSE 2017   Algorithm
Question 1 Explanation:
 Question 2
Language $L_{1}$ is polynomial time reducible to language $L_{2}$. Language $L_{3}$ is polynomial time reducible to $L_{2}$, which in turn is polynomial time reducible to language $L_{4}$. Which of the following is/are true?
I. if $L_{4} \in P$, then $L_{2} \in P$
II. if $L_{1} \in P$ or $L_{3} \in P$, then $L_{2} \in P$
III. $L_{1} \in P$, if and only if $L_{3} \in P$
IV. if $L_{4} \in P$, then $L_{1} \in P$ and $L_{3} \in P$
 A II only B III only C I and IV only D I only
GATE CSE 2015 SET-3   Algorithm
Question 2 Explanation:

 Question 3
Consider two decision problems Q1,Q2 such that Q1 reduces in polynomial time to 3-SAT and 3-SAT reduces in polynomial time to Q2. Then which one of the following is consistent with the above statement?
 A Q1 is in NP, Q2 is NP hard. B Q2 is in NP, Q1 is NP hard C Both Q1 and Q2 are in NP . D Both Q1 and Q2 are NP hard
GATE CSE 2015 SET-2   Algorithm
Question 3 Explanation:
 Question 4
Consider the decision problem 2CNFSAT defined as follows:
{ $\phi$ | $\phi$ is a satisfiable propositional formula in CNF with at most two literal per clause}
For example, $\phi =(x_{1} \vee x_{2}) \wedge (x_{1} \vee \bar{x_{3}}) \wedge (x_{2} \vee x_{4})$ is a Boolean formula and it is in 2CNFSAT.
The decision problem 2CNFSAT is
 A NP-Complete B solvable in polynomial time by reduction to directed graph reachability. C solvable in constant time since any input instance is satisfiable D NP-hard, but not NP-complete
GATE CSE 2014 SET-3   Algorithm
Question 4 Explanation:
 Question 5
Suppose a polynomial time algorithm is discovered that correctly computes the largest clique in a given graph. In this scenario, which one of the following represents the correct Venn diagram of the complexity classes P, NP and NP Complete (NPC)? A A B B C C D D
GATE CSE 2014 SET-1   Algorithm
Question 5 Explanation:

There are 5 questions to complete.