Why it matters
The answer would reshape complexity theory and clarify the limits of efficient computation, with consequences for optimization, proof search, cryptography, and algorithm design.
Context and known approaches
P contains decision problems solvable in polynomial time. NP contains decision problems whose proposed solutions can be verified in polynomial time.
Formal boundary
Determine whether the complexity classes P and NP are equal.
Known partial results
- Multiple broad proof-strategy barriers are known; see the cited dossier milestones.
Equivalent formulations
- Many NP-complete problems give equivalent yes/no consequences under polynomial reductions.
Source trail
- Clay Mathematics Institute — Official Millennium problem overview.
- Wikipedia — Definitions, history, and references.
- Wikipedia list subsection — Placement in the accepted revision-pinned living-list snapshot.