When considering P, NP, NP complete, and NP hard

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Question: When considering P, NP, NP complete, and NP hard

Option 1: 'NP hard is a subset of NP complete' The statement mixes two related but distinct concepts. NP-hard problems are at least as hard as the hardest problems in NP, but they do not have to be in NP themselves. NP-complete problems are those problems that are in NP and are NP-hard. Therefore, claiming NP-hard is a subset of NP-complete is incorrect because many NP-hard problems lie outside NP and thus cannot be contained within the set of NP-complete problems.

Option 2: 'P = NP' This is a famous open question in computer science. While it is a widely studied hypothesis, there is no proven proof that P equals NP or that P does not equal NP (as of now). Treating this as a definite equality is therefore incorrect in the current state of knowledge, since the formal proof or disproof has not been established.

Option 3: 'P is a subset of NP' This statement aligns with standard complexity theory. Every problem that can be solved in deterministic polynomial time (class P) can also be verified in nondeterministic polynomial time (class NP) because a solution can be produced and checked efficiently. Hence P is contained within NP.

Option 4: 'NP is the sum of NP hard and NP complete' The phrasing here is not standard terminology. Complexity classes are not formed by summing other classes; NP comprises all problems whose solutions can be verified in polynomial time, and NP-hard or NP-complete are classifications based on hardness and membership criteria. Combining NP-hard and NP-complete as a sum does not reflect the formal relationships among these classes.

INSY-7100-001 (Fall 2025) Multiobjective Optimization and Constraint Handling

When considering P, NP, NP complete, and NP hard

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