GATE CS 2014 Set 1 — Question 48
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Algorithms → Complexity Theory → P, NP, NP-Complete
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Question
Suppose a polynomial time algorithm is discovered that correctly computes the largest clique in a given graph. In this scenario, which one of the following represents the correct Venn diagram of the complexity classes P, NP and NP Complete (NPC)?
Correct answer
(D) [figure]
Solution
Finding a largest clique solves the CLIQUE decision problem, which is NP-complete. A polynomial-time algorithm for an NP-complete problem gives a polynomial-time algorithm for every problem in NP (reduce to CLIQUE, then solve it), so .If , then every problem in NP is polynomial-time reducible to every other non-trivial problem in NP, so every such problem is NP-complete. The official answer takes this to mean that P, NP and NPC coincide, which is diagram (D).
- (A) shows P strictly inside NP, which is wrong once .
- (B) shows P and NP as different overlapping sets, also wrong once .
- (C) shows with NPC as a proper subset.
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