Why it matters

NP-Hard broader than NPC. Understanding shapes hardness classification.

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The architecture

Every NP problem reduces to it.

Not required to be in NP itself.

NP-Hard classEvery NP reducesto problemNot required NPmay be harderExamplesTSP-opt, haltingTSP decision NPC; TSP optimization NP-hard; halting NP-hard but undecidable
NP-Hard.
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How it works end to end

NPC = NP-Hard AND in NP.

Halting Problem NP-hard (undecidable).

Optimization variants often NP-hard.