Assume that P and NP are different i.e. P! = NP then for the expression NP-Complete ∩ P = ? Which among the following is correct ?

Question

Question

This question was previously asked in

NIELIT Scientific Assistant CS 5 Dec 2021 Official Paper

Answer 1

Correct Answer: Option 2) ∅

Key Points:

NP-Complete problems are those that are:
- In NP (verifiable in polynomial time), and
- As hard as any problem in NP (i.e., every NP problem reduces to it in polynomial time).
If a problem is both in P and NP-Complete, then all NP problems can be solved in polynomial time, i.e., P = NP.
But it is given that P ≠ NP, so no NP-Complete problem can be in P.

Therefore, the intersection of P and NP-Complete is empty (∅).

Additional Information:

NP-Hard: Includes problems at least as hard as NP problems but not necessarily in NP (not even decidable).
P: Problems solvable in polynomial time.
NP: Problems whose solutions can be verified in polynomial time.
NP-Complete: Subset of NP problems that are also NP-Hard.

Hence: If P ≠ NP, no NP-Complete problem lies in P ⇒ NP-Complete ∩ P = ∅