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NP: problems whose solution can be e
f
ciently (poly-time) verifed.
Ex: Compositeness: Given n-bit number x, in x composite?
“Hint”: a pair y,z > 1 such that yz = x.
NP-Complete: a problem X which is in NP and every other problem
Y in NP has a poly-time reduction FROM Y TO X.
Ex. Ind. Set, 3 SAT.
Alt de±: a problem X which is in NP and there exists NPC
problem Y which has a poly-time reduction ±rom Y to X.
2 easy reductions ±rom ind. set.
(1) CLIQUE: Given undirected graph G = (V, E) and a number k >
0.
Output “yes” i± G has a subgraph H with k vertices and every
2 vertices o± H are joined by an edge.
(H is a “k-clique”.)
Take (G, k) an instance o± IND. SET.
Construct G
̅
which has
same vertex set and (u,v)
∈
E(G
̅
) <=> (u,v)
∉
E(G).
(1) Reduction runs in poly-time.
YES! O(n
2
)
(2) I± G has k-ind set, G
̅
has a k-clique.
(3) I± G has no k-ind set, G

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