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Difference between NP-hard and NP-complete problems

CategoryNP-hard ProblemsNP-complete Problems
DefinitionA problem that is at least as hard as the hardest problems in NP.A problem that belongs to both NP and is NP-hard.
Polynomial-time ReductionPolynomial-time reduction from any NP problem to an NP-hard problem.Polynomial-time reduction from any NP problem to an NP-complete problem.
Membership in NPNot necessarily in NP.Must be in NP.
SolvabilityMay or may not be solvable in polynomial time.Unlikely to have a polynomial-time algorithm; no known polynomial-time algorithm exists.
ExamplesTraveling Salesman Problem (TSP), 3-SAT, Vertex Cover.Boolean Satisfiability (SAT), Knapsack Problem, Hamiltonian Cycle.
ImportanceServes as a benchmark for the complexity of problems.Identifies the hardest problems in NP and serves as a basis for many other complexity results.
RelationshipAll NP-complete problems are also NP-hard.Not all NP-hard problems are NP-complete. Some NP-hard problems may not be in NP.
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