Question
Download Solution PDFAssume that P and NP are different i.e. P! = NP then for the expression NP-Complete ∩ P = ? Which among the following is correct ?
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFThe correct answer is NP-Hard.
Key Points
- Assuming P ≠ NP implies that problems in P are not the same as NP-Complete problems.
- NP-Complete problems are the hardest problems in NP, in the sense that if any NP-Complete problem can be solved in polynomial time, then every problem in NP can be solved in polynomial time.
- If a problem is both in NP-Complete and in P, it would imply P = NP, which contradicts our assumption.
- Therefore, the intersection of NP-Complete and P must be empty under the assumption that P ≠ NP.
- So, NP-Complete ∩ P = ∅ (the empty set).
Additional Information
- NP-Hard problems are at least as hard as the hardest problems in NP, but they do not necessarily have to be in NP.
- If a problem is NP-Hard, it means solving it in polynomial time would allow all NP problems to be solved in polynomial time.
- The concept of NP-Hard is crucial in the study of computational complexity theory, as it helps in understanding the limits of efficient computation.
- In practical terms, knowing a problem is NP-Hard helps in setting realistic expectations for finding efficient algorithms for it.
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