Worked Examples: NP-Completeness

These examples are designed to step through the logical process of applying the algorithmic concepts.

Example 1: Class P

❓ Problem: Definition.

💡 Solution: Solvable in Poly-time.

Example 2: Class NP

❓ Problem: Definition.

💡 Solution: Verifiable in Poly-time (Nondeterministic Poly time).

Example 3: Reduction Direction

❓ Problem: If $A \le_p B$ and B in P?

💡 Solution: A is in P.

Example 4: Reduction Direction 2

❓ Problem: If $A \le_p B$ and A is Hard?

💡 Solution: B is Hard.

Example 5: SAT

❓ Problem: Input/Output?

💡 Solution: Boolean formula. Yes if assignment makes True.

Example 6: NP-Hard vs NPC

❓ Problem: Difference?

💡 Solution: NPC must be in NP. NP-Hard doesn't (Halting Problem).