3/24/08  NPcomplete sequencing pro.
..
NP: problems whose solution can be e
f
ciently (polytime) verifed.
Ex: Compositeness: Given nbit number x, in x composite?
“Hint”: a pair y,z > 1 such that yz = x.
NPComplete: a problem X which is in NP and every other problem
Y in NP has a polytime 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 polytime 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 “kclique”.)
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 polytime.
YES! O(n
2
)
(2) I± G has kind set, G
̅
has a kclique.
(3) I± G has no kind set, G
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 Spring '08
 KLEINBERG
 Algorithms, Graph Theory, vertex cover, Glossary of graph theory, undirected graph, polytime

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