Key differences between P, NP, NP-complete, and NP-hard

- Verifiable and solvable in polynomial time.
- A subset of NP problems. | Sorting (e.g., Merge Sort), Searching | | NP (Nondeterministic Polynomial) | Problems whose solutions can be verified in polynomial time by a deterministic algorithm. | - Solution verification is efficient (polynomial time).
- Includes all problems in P.
- May not be efficiently solvable. | Boolean Satisfiability (SAT), Graph Coloring | | NP-complete | Problems that are both in NP and as hard as the hardest problems in NP (NP-hard and in NP). | - If any NP-complete problem is solved in polynomial time, all NP problems can be solved in polynomial time.
- Requires verification and reduction properties. | 3-SAT, Traveling Salesman (decision version) | | NP-hard | Problems as hard as NP-complete problems but not necessarily in NP (not required to be verifiable in polynomial time). | - May not even be solvable or verifiable in polynomial time.
- Includes optimization problems and undecidable problems. | Halting Problem, Bin Packing Problem |

Key Differences:

1. Problem Solving:

   • P: Problems can be both solved and verified in polynomial time.

   • NP: Problems may not be solved in polynomial time but can be verified in polynomial time.

   • NP-complete: Problems are the hardest in NP and are solvable if any NP problem is solved in polynomial time.

   • NP-hard: Problems are as hard as NP-complete but not necessarily in NP (i.e., they might not even have a verifiable solution in polynomial time).

2. Inclusion:

   • P: Subset of NP.

   • NP-complete: Subset of NP-hard.

   • NP-hard: Not necessarily a subset of NP.

3. Implication:

   • Solving any NP-complete problem in polynomial time implies P=NP.

   • Solving an NP-hard problem does not necessarily affect P=NP, as NP-hard problems might not be verifiable.

Visual Relationship:

• P ⊆ NP.

• NP-complete ⊆ NP.

• NP-complete ⊆ NP-hard.

• NP-hard includes problems that might not belong to NP.