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NP-Completeness Proofs

Lemma 34.8

If L is a language such that L' \(\leq_P\) L for some L' $\in$ NPC, then L is NP-Hard. If also L $\in$ NP, then L $\in$ NPC.

Strategy for proving L $\in$ NPC

1.
Prove L $\in$ NP (poly-time verifiable)
2.
Select L' $\in$ NPC
3.
Describe poly-time algorithm computing a function f that maps instances of L' to instances of L
4.
Prove that x $\in$ L' iff f(x) $\in$ L for all x $\in$ {0,1}*.
Note: Showing L' \(\leq_P\) spec(L) implies L' \(\leq_P\) L.


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