Why it matters

Recognizing NP-completeness saves effort. Understanding shapes approach to hard problems.

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The architecture

NP: verifiable in poly time.

NP-complete: NP + all NP problems reduce to it.

Complexity hierarchyVerification polyNPReduction from NP-CNP-hardBoth = NP-completehardest in NPClassic NP-C: SAT, TSP, knapsack, graph coloring, clique
NP-C definition.
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How it works end to end

Reduction: transform problem A to B in poly time. If B solvable poly, so is A.

Approaches for NP-hard: approximation algorithms, heuristics, SAT/ILP solvers, special-case algorithms.

Classic NP-C: SAT, TSP, knapsack, graph coloring.