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NP-complete (Definition)

A problem $ \pi\in\mathcal{NP}$ is $ \mathcal{NP}$ complete if for any $ \pi^\prime\in\mathcal{NP}$ there is a Cook reduction of $ \pi^\prime$ to $ \pi$. Hence if $ \pi\in\mathcal{P}$ then every $ \mathcal{NP}$ problem would be in $ \mathcal{P}$. A slightly stronger definition requires a Karp reduction or Karp reduction of corresponding decision problems as appropriate.

A search problem $ R$ is $ \mathcal{NP}$ hard if for any $ R^\prime\in\mathcal{NP}$ there is a Levin reduction of $ R^\prime$ to $ R$.



"NP-complete" is owned by Henry.
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Other names:  NP complete, NP hard
Also defines:  NP-hard
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Cross-references: Levin reduction, search problem, decision problems, Karp reduction, Cook reduction
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This is version 1 of NP-complete, born on 2002-09-06.
Object id is 3429, canonical name is NPComplete.
Accessed 7799 times total.

Classification:
AMS MSC: 68Q15 (Computer science :: Theory of computing :: Complexity classes )
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