Why it matters

Karp reductions prove NP-hardness. Understanding shapes hardness proofs.

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The architecture

Transform A instance to B instance in poly time.

Preserve yes/no answer.

If B poly then A poly.

Karp reduction structureA instanceinputPoly transformto B instanceB instanceyes/no preservedIf A NP-hard + A reduces to B, then B NP-hard (poly reduces preserved)
Karp reduction.
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How it works end to end

Karp 1972.

Notation: A <=_p B.

Transitive: chain reductions.

Foundation for NP-hardness.