additional np-complete problems
DESCRIPTION
Additional NP-Complete Problems. Lecture 40 Section 7.5 Mon, Dec 3, 2007. CLIQUE is NP-Complete. We have already seen that 3SAT is NP-complete and 3SAT can be reduced to CLIQUE. Therefore, CLIQUE is NP-complete. VERTEX-COVER is NP-complete. - PowerPoint PPT PresentationTRANSCRIPT
Additional NP-Complete Problems
Lecture 40Section 7.5
Mon, Dec 3, 2007
CLIQUE is NP-Complete
• We have already seen that • 3SAT is NP-complete and• 3SAT can be reduced to CLIQUE.
• Therefore, CLIQUE is NP-complete.
VERTEX-COVER is NP-complete
• The Vertex-cover problem: Given a graph G and an integer k, does there exist a set S of k vertices such that every edge of G has at least one endpoint in S?
VERTEX-COVER is NP-complete
• We will now reduce CLIQUE to VERTEX-COVER to show that VERTEX-COVER is NP-complete.
• We have already shown that VERTEX-COVER is in NP.
VERTEX-COVER is NP-complete
• Let G, k be an instance of CLIQUE.
• That is, G is a graph, k is an integer, and the question is whether G has a clique of size k.
VERTEX-COVER is NP-complete
• Create the graph G as follows.• V(G ) = V(G)• e is an edge of G if and only if e is
not an edge of G.
• G is the complement of G.• We claim that G has a clique of
size k if and only if G has a vertex cover of size n – k.
Example
G
Example
G G
Example
G G
Example
G G
k = 3 n – k = 5
Example
G G
k = 3 n – k = 5
Example
G G
k = 3 n – k = 5
Example
G G
k = 3 n – k = 5
Example
G G
k = 3 n – k = 5
Example
G G
k = 3 n – k = 5
Example
G G
k = 3 n – k = 5
Example
G G
k = 3 n – k = 5
Example
G G
k = 3 n – k = 5
VERTEX-COVER is NP-complete
• Proof• Let S be a vertex cover of G.• Let T = V – S.• We claim that T is a clique of G.• Let vertices i and j be in T.• Then i and j are not in S.• Therefore, there is no edge in G from
i to j because S is a vertex cover.
VERTEX-COVER is NP-complete
• Then there is an edge from i to j in G.• Therefore, T is a clique in G.• Therefore, “yes” to vertex cover
implies “yes” to clique. vertex cover S of n – k in G
clique T of k in G.
VERTEX-COVER is NP-complete
• Now we must show that( vertex cover S of n – k in G)
( clique T of k in G).• But this is the same as showing that
clique T of k in G vertex cover S of n – k in G.
VERTEX-COVER is NP-complete
• So now suppose that T is a clique of size k in G.
• Let S = V – T.• We claim that S is a vertex cover of
G.• Let e be an edge in G.• Then e must have at least one
endpoint in S.
VERTEX-COVER is NP-complete
• Therefore, S is a vertex cover of G.• So “yes” to clique implies “yes” to
vertex cover.• Therefore, “no” to vertex cover
implies “no” to clique.