Class NP:

• Definition: NP is the class of decision problems for which a proposed solution can be verified in polynomial time.

Class P:

• Definition: P is the class of decision problems that can be solved in polynomial time.

NP-Complete Problems:

• Definition: NP-complete problems are the hardest problems in NP. They are in NP and every problem in NP can be reduced to them in polynomial time.
• Implication: If a polynomial-time solution exists for one NP-complete problem, then all problems in NP can be solved in polynomial time (P = NP).

Polynomial-Time Reduction:

• Definition: If a problem A can be transformed into problem B in polynomial time, and B is known to be NP-complete, then A is also NP-complete.

Significance:

• Importance: NP-complete problems are central to understanding computational complexity. If it’s proven that P ≠ NP, NP-complete problems are inherently difficult to solve efficiently.