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Ram and Shyam have been asked to show that a cartain problem II is NP-complete. Ram shows a polynomial time reduction from the 3-SAT problem to II and Shyam shows a polynomial time reduction from II to 3-SAT. Which of the following can be inferred from these reductions?
- a)II is NP-hard but not NP-comp!ete
- b)II is NP-complete
- c)II is in NP, but is not NP-complete
- d)II is neither NP-hard, nor in NP
Correct answer is option 'B'. Can you explain this answer?
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Verified Answer
Ram and Shyam have been asked to show that a cartain problem II is NP-...
Since 3-SAT problem is NP-complete and 3-SAT is polynomial time reducible to π hence π is is also NP-complete.
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Ram and Shyam have been asked to show that a cartain problem II is NP-...
Reductions and NP-Completeness
To understand the answer to this question, we need to have a clear understanding of reductions and NP-completeness.
- Reductions: In computational complexity theory, a reduction is a way to solve one problem using the solution of another problem. A reduction from problem A to problem B means that an algorithm for solving problem B can be used to solve problem A. If a reduction can be done in polynomial time, it is called a polynomial-time reduction.
- NP-Completeness: A problem is said to be NP-complete if it is in the class of NP (nondeterministic polynomial time) problems and every problem in NP can be reduced to it in polynomial time. In simpler terms, if a problem is NP-complete, it is one of the hardest problems in NP.
Explanation of the Given Scenario
Ram shows a polynomial time reduction from the 3-SAT problem to II, while Shyam shows a polynomial time reduction from II to 3-SAT. Let's analyze the implications of these reductions.
- If Ram can reduce 3-SAT (which is a known NP-complete problem) to II in polynomial time, it means that II is at least as hard as 3-SAT. In other words, II is NP-hard.
- If Shyam can reduce II to 3-SAT in polynomial time, it means that II can be solved using the solution of 3-SAT. Since 3-SAT is NP-complete, if II can be reduced to 3-SAT in polynomial time, it implies that II is in NP.
Inference from the Reductions
Combining the implications of both reductions, we can infer the following:
- II is NP-hard because it can be reduced to 3-SAT in polynomial time (as shown by Ram).
- II is in NP because it can be reduced from II to 3-SAT in polynomial time (as shown by Shyam).
Conclusion
Based on the reductions provided by Ram and Shyam, we can conclude that:
- II is NP-hard (as it can be reduced from the NP-complete problem 3-SAT).
- II is in NP (as it can be reduced to the NP-complete problem 3-SAT).
Hence, the correct answer is option 'B': II is NP-complete.
To understand the answer to this question, we need to have a clear understanding of reductions and NP-completeness.
- Reductions: In computational complexity theory, a reduction is a way to solve one problem using the solution of another problem. A reduction from problem A to problem B means that an algorithm for solving problem B can be used to solve problem A. If a reduction can be done in polynomial time, it is called a polynomial-time reduction.
- NP-Completeness: A problem is said to be NP-complete if it is in the class of NP (nondeterministic polynomial time) problems and every problem in NP can be reduced to it in polynomial time. In simpler terms, if a problem is NP-complete, it is one of the hardest problems in NP.
Explanation of the Given Scenario
Ram shows a polynomial time reduction from the 3-SAT problem to II, while Shyam shows a polynomial time reduction from II to 3-SAT. Let's analyze the implications of these reductions.
- If Ram can reduce 3-SAT (which is a known NP-complete problem) to II in polynomial time, it means that II is at least as hard as 3-SAT. In other words, II is NP-hard.
- If Shyam can reduce II to 3-SAT in polynomial time, it means that II can be solved using the solution of 3-SAT. Since 3-SAT is NP-complete, if II can be reduced to 3-SAT in polynomial time, it implies that II is in NP.
Inference from the Reductions
Combining the implications of both reductions, we can infer the following:
- II is NP-hard because it can be reduced to 3-SAT in polynomial time (as shown by Ram).
- II is in NP because it can be reduced from II to 3-SAT in polynomial time (as shown by Shyam).
Conclusion
Based on the reductions provided by Ram and Shyam, we can conclude that:
- II is NP-hard (as it can be reduced from the NP-complete problem 3-SAT).
- II is in NP (as it can be reduced to the NP-complete problem 3-SAT).
Hence, the correct answer is option 'B': II is NP-complete.
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Ram and Shyam have been asked to show that a cartain problem II is NP-complete. Ram shows a polynomial time reduction from the 3-SAT problem to II and Shyam shows a polynomial time reduction from II to 3-SAT. Which of the following can be inferred from these reductions?a)II is NP-hard but not NP-comp!eteb)II is NP-completec)II is in NP, but is not NP-completed)II is neither NP-hard, nor in NPCorrect answer is option 'B'. Can you explain this answer?
Question Description
Ram and Shyam have been asked to show that a cartain problem II is NP-complete. Ram shows a polynomial time reduction from the 3-SAT problem to II and Shyam shows a polynomial time reduction from II to 3-SAT. Which of the following can be inferred from these reductions?a)II is NP-hard but not NP-comp!eteb)II is NP-completec)II is in NP, but is not NP-completed)II is neither NP-hard, nor in NPCorrect answer is option 'B'. Can you explain this answer? for GATE 2024 is part of GATE preparation. The Question and answers have been prepared according to the GATE exam syllabus. Information about Ram and Shyam have been asked to show that a cartain problem II is NP-complete. Ram shows a polynomial time reduction from the 3-SAT problem to II and Shyam shows a polynomial time reduction from II to 3-SAT. Which of the following can be inferred from these reductions?a)II is NP-hard but not NP-comp!eteb)II is NP-completec)II is in NP, but is not NP-completed)II is neither NP-hard, nor in NPCorrect answer is option 'B'. Can you explain this answer? covers all topics & solutions for GATE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Ram and Shyam have been asked to show that a cartain problem II is NP-complete. Ram shows a polynomial time reduction from the 3-SAT problem to II and Shyam shows a polynomial time reduction from II to 3-SAT. Which of the following can be inferred from these reductions?a)II is NP-hard but not NP-comp!eteb)II is NP-completec)II is in NP, but is not NP-completed)II is neither NP-hard, nor in NPCorrect answer is option 'B'. Can you explain this answer?.
Ram and Shyam have been asked to show that a cartain problem II is NP-complete. Ram shows a polynomial time reduction from the 3-SAT problem to II and Shyam shows a polynomial time reduction from II to 3-SAT. Which of the following can be inferred from these reductions?a)II is NP-hard but not NP-comp!eteb)II is NP-completec)II is in NP, but is not NP-completed)II is neither NP-hard, nor in NPCorrect answer is option 'B'. Can you explain this answer? for GATE 2024 is part of GATE preparation. The Question and answers have been prepared according to the GATE exam syllabus. Information about Ram and Shyam have been asked to show that a cartain problem II is NP-complete. Ram shows a polynomial time reduction from the 3-SAT problem to II and Shyam shows a polynomial time reduction from II to 3-SAT. Which of the following can be inferred from these reductions?a)II is NP-hard but not NP-comp!eteb)II is NP-completec)II is in NP, but is not NP-completed)II is neither NP-hard, nor in NPCorrect answer is option 'B'. Can you explain this answer? covers all topics & solutions for GATE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Ram and Shyam have been asked to show that a cartain problem II is NP-complete. Ram shows a polynomial time reduction from the 3-SAT problem to II and Shyam shows a polynomial time reduction from II to 3-SAT. Which of the following can be inferred from these reductions?a)II is NP-hard but not NP-comp!eteb)II is NP-completec)II is in NP, but is not NP-completed)II is neither NP-hard, nor in NPCorrect answer is option 'B'. Can you explain this answer?.
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