Choose the correct statement(s):
A. A problem which is NP-Complete will have the property that it can be solved in polynomial time iff all other NP-complete problems can also be solved in polynomial time.
B. All NP-complete problems are NP-hard problems.
C. If an NP-hard problem can be solved in polynomial time, then all NP-complete problems can be solved in polynomial time.
D. All NP-hard problems are not NP-complete.
Choose the correct answer from the options given below:

Correct option is D

The given statements revolve around NP-Complete and NP-Hard problems. Let's break down each statement:
A. A problem which is NP-Complete will have the property that it can be solved in polynomial time iff all other NP-complete problems can also be solved in polynomial time.
This statement is true. The defining characteristic of NP-Complete problems is that if one NP-Complete problem can be solved in polynomial time, then all NP-Complete problems can be solved in polynomial time, making them solvable in polynomial time collectively.
B. All NP-complete problems are NP-hard problems.
This statement is true. By definition, NP-Complete problems are a subset of NP-Hard problems. NP-Hard problems are those that are at least as hard as any NP problem, and NP-Complete problems are the hardest NP problems.
C. If an NP-hard problem can be solved in polynomial time, then all NP-complete problems can be solved in polynomial time.
This statement is true. If any NP-Hard problem is solvable in polynomial time, then all NP-Complete problems (a subset of NP-Hard problems) must also be solvable in polynomial time. This is because the solution to any NP-Hard problem can be used to solve NP-Complete problems in polynomial time.
D. All NP-hard problems are not NP-complete.
This statement is true. Not all NP-Hard problems are NP-Complete. NP-Complete problems are a subset of NP-Hard problems, specifically those that are in NP as well. There exist NP-Hard problems that are not in NP and, hence, not NP-Complete.
Information Booster:
1. NP-Complete:
o NP-Complete problems are the hardest problems in the class NP (nondeterministic polynomial time), where all problems in NP can be reduced to them in polynomial time.
o They are characterized by two properties: being in NP and being as hard as any problem in NP (i.e., every problem in NP can be reduced to them).
2. NP-Hard:
o NP-Hard problems include all problems that are at least as hard as the hardest problems in NP. They may or may not be in NP.
o NP-Hard includes problems that are more general and possibly harder than NP problems.
3. Polynomial Time: If any NP-Complete or NP-Hard problem is solvable in polynomial time, it implies that the entire class NP can be solved in polynomial time, thus P = NP (a major unsolved question in computer science).
4. Relationship Between NP-Hard and NP-Complete: NP-Complete problems are those that are both in NP and NP-Hard. If a problem is NP-Hard but not in NP, it is not NP-Complete.
Additional Knowledge:
• P = NP: The question of whether P = NP is still one of the major open problems in computer science. If P = NP, it would mean that all NP problems, including NP-Complete, can be solved in polynomial time.
• NP-Hard Problems: An NP-Hard problem can be as difficult as, or harder than, the hardest problem in NP. However, they are not guaranteed to be solvable in polynomial time, nor are they required to be in NP.