Complexity Theory

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## TL;DR

Computational complexity classifies problems by the resources (time, memory) required to solve them. P: solvable in polynomial time by deterministic Turing machine. NP: verifiable in polynomial time. NP-complete: hardest problems in NP (SAT, TSP, knapsack). P vs NP: the greatest open problem in computer science.

## Core Explanation

P ⊆ NP ⊆ PSPACE ⊆ EXPTIME. NP-complete problems: if you can solve one efficiently, you can solve all (polynomial-time reduction). Cook-Levin Theorem: SAT is NP-complete. Consequences of P=NP: cryptography would break (factoring in P), optimization becomes easy. Most computer scientists believe P ≠ NP (though unproven).

## Further Reading

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