GATE CS 2015 Set 2 — Question 26
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Algorithms → Complexity Theory → Polynomial Reductions
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Question
Consider two decision problems such that reduces in polynomial time to 3-SAT and 3-SAT reduces in polynomial time to . Then which one of the following is consistent with the above statement?
Correct answer
(A) Q₁ is in NP, Q₂ is NP hard.
Solution
Given:
From (2), since 3-SAT is NP-Hard, must be NP-Hard (if an NP-Hard problem reduces to a problem, that problem is NP-Hard).Thus, is in NP and is NP-Hard.
1.
2.
From (1), since 3-SAT is in NP, must be in NP (if a problem reduces to an NP problem, it is in NP).From (2), since 3-SAT is NP-Hard, must be NP-Hard (if an NP-Hard problem reduces to a problem, that problem is NP-Hard).Thus, is in NP and is NP-Hard.
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