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Subset Sum Problem

SUBSET-SUM = {$\langle$S, t$\rangle$$\mid$ there exists S' $\subseteq$ S $\subset$ N such that \(\sum_{s \in S'} s\) = t $\in$ N},
N = set of natural numbers

Theorem 34.15: SUBSET-SUM $\in$ NPC

Proof:

1.
SUBSET-SUM $\in$ NP. Just add up elements of S' and compare sum to t.
2.
L' = 3-CNF-SAT
3.
3-CNF-SAT \(\leq_P\) SUBSET-SUM
4.
x $\in$ 3-CNF-SAT $\leftrightarrow$ f(x) $\in$ SS
Proof is complex.

Alternate proof: Reduce from VERTEX-COVER (1st edition).


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