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Theorem 36.12: VC
NPC
Proof Sketch:
1.
VC
NP
Given V', check
V'
= k, and for each edge (u,v)
E, check that either u
V' or v
V'.
2.
L' = CLIQUE
3.
CLIQUE
VC
If graph G = (V, E) has clique V', then graph
has vertex cover V - V'.
= (V,
) is the
complement
of G = (V,E), where
Reduction: G
(poly-time)
4.
x
CLIQUE(G) = V'
f(x)
VC(
) = V - V' ( |
V
' | =
k
)
Every edge (u,v)
implies (u,v)
E, thus at least one of u and v
V'. Thus, at least one of u,v belongs to V - V', which means edge (u,v) is covered by V - V'. Similar argument for other direction.
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