# Top MCQs on NP Complete Complexity with Answers

Question 1

Let S be an NP-complete problem and Q and R be two other problems not known to be in NP. Q is polynomial time reducible to S and S is polynomial-time reducible to R. Which one of the following statements is true? (GATE CS 2006)
• R is NP-complete
• R is NP-hard
• Q is NP-complete
• Q is NP-hard

Question 2

Let X be a problem that belongs to the class NP. Then which one of the following is TRUE?
• There is no polynomial time algorithm for X.
• If X can be solved deterministically in polynomial time, then P = NP.
• If X is NP-hard, then it is NP-complete.
• X may be undecidable.

Question 3

The problem 3-SAT and 2-SAT are

• both in P

• both NP complete

• NP-complete and in P respectively

• undecidable and NP-complete respectively

Question 4

Which of the following statements are TRUE? (1) The problem of determining whether there exists a cycle in an undirected graph is in P. (2) The problem of determining whether there exists a cycle in an undirected graph is in NP. (3) If a problem A is NP-Complete, there exists a non-deterministic polynomial time algorithm to solve A.
• 1, 2 and 3
• 1 and 3
• 2 and 3
• 1 and 2

Question 5

Which of the following statements are TRUE?
```1. The problem of determining whether there exists
a cycle in an undirected graph is in P.
2. The problem of determining whether there exists
a cycle in an undirected graph is in NP.
3. If a problem A is NP-Complete, there exists a
non-deterministic polynomial time algorithm to solve A.
```
• 1, 2 and 3
• 1 and 2 only
• 2 and 3 only
• 1 and 3 only

Question 6

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
• B
• C
• D

Question 7

Consider the decision problem 2CNFSAT defined as follows:
• NP-Complete.
• solvable in polynomial time by reduction to directed graph reachability.
• solvable in constant time since any input instance is satisfiable.
• NP-hard, but not NP-complete.

Question 8

Consider the following two problems on undirected graphs
```α : Given G(V, E), does G have an independent set of size | V | - 4?
β : Given G(V, E), does G have an independent set of size 5?
```
Which one of the following is TRUE?
• α is in P and β is NP-complete
• α is NP-complete and β is in P
• Both α and β are NP-complete
• Both α and β are in P

Question 9

Ram and Shyam have been asked to show that a certain problem Π is NP-complete. Ram shows a polynomial time reduction from the 3-SAT problem to Π, and Shyam shows a polynomial time reduction from Π to 3-SAT. Which of the following can be inferred from these reductions ?
• Π is NP-hard but not NP-complete
• Π is in NP, but is not NP-complete
• Π is NP-complete
• Π is neither NP-hard, nor in NP

Question 10

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?
• Q1 is in NP, Q2 is NP hard
• Q2 is in NP, Q1 is NP hard
• Both Q1 and Q2 are in NP
• Both Q1 and Q2 are in NP hard

There are 20 questions to complete.

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