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
The Largest Clique problem is an NP-Complete (NPC) problem. If a polynomial-time algorithm is discovered for an NPC problem, it implies that every problem in NP can be reduced to it and then solved in polynomial time. Therefore, .If , then the set P and the set NP are identical. The set of NP-Complete problems (NPC) consists of the "hardest" problems in NP. If , then all problems in NP (except for trivial ones like the empty language and ) are NP-Complete. Thus, NPC is a proper subset of (containing all non-trivial problems).Option (C) correctly depicts a single circle for with as a subset inside it. Option (D) suggests , which is technically incorrect because trivial problems in P are not NP-Complete.
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