The.Algorithm.Design.Manual.Springer-Verlag.1998
The.Algorithm.Design.Manual.Springer-Verlag.1998 The.Algorithm.Design.Manual.Springer-Verlag.1998
Lecture 21 - vertex cover Every vertex cover must contain n first stage vertices and 2C second stage vertices. Let the first stage vertices define the truth assignment. To give the cover, at least one cross-edge must be covered, so the truth assignment satisfies. For a cover to have N+2C vertices, all the cross edges must be incident on a selected vertex. Let the N selected vertices from the first stage coorespond to TRUE literals. If there is a satisfying truth assignment, that means at least one of the three cross edges from each triangle is incident on a TRUE vertex. By adding the other two vertices to the cover, we cover all edges associated with the clause. Every SAT defines a cover and Every Cover Truth values for the SAT! Example: , . Listen To Part 25-1 Starting from the Right Problem As you can see, the reductions can be very clever and very complicated. While theoretically any NPcomplete problem can be reduced to any other one, choosing the correct one makes finding a reduction much easier. As you can see, the reductions can be very clever and complicated. While theoretically any NP-complete problem will do, choosing the correct one can make it much easier. Instance: A graph G=(V,E) and integer . file:///E|/LEC/LECTUR16/NODE21.HTM (3 of 7) [19/1/2003 1:35:28] Maximum Clique
Lecture 21 - vertex cover Question: Does the graph contain a clique of j vertices, ie. is there a subset of v of size j such that every pair of vertices in the subset defines an edge of G? Example: this graph contains a clique of size 5. Listen To Part 25-2 When talking about graph problems, it is most natural to work from a graph problem - the only NPcomplete one we have is vertex cover! Theorem: Clique is NP-complete Proof: If you take a graph and find its vertex cover, the remaining vertices form an independent set, meaning there are no edges between any two vertices in the independent set, for if there were such an edge the rest of the vertices could not be a vertex cover. Clearly the smallest vertex cover gives the biggest independent set, and so the problems are equivallent - Delete the subset of vertices in one from the total set of vertices to get the order! Thus finding the maximum independent set must be NP-complete! Listen To Part 25-3 In an independent set, there are no edges between two vertices. In a clique, there are always between two vertices. Thus if we complement a graph (have an edge iff there was no edge in the original graph), a clique becomes an independent set and an independent set becomes a Clique! file:///E|/LEC/LECTUR16/NODE21.HTM (4 of 7) [19/1/2003 1:35:28]
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Lecture 21 - vertex cover<br />
Question: Does the graph contain a clique of j vertices, ie. is there a subset of v of size j such that every<br />
pair of vertices in the subset defines an edge of G?<br />
Example: this graph contains a clique of size 5.<br />
Listen To Part 25-2<br />
When talking about graph problems, it is most natural to work from a graph problem - the only NPcomplete<br />
one we have is vertex cover!<br />
<strong>The</strong>orem: Clique is NP-complete<br />
Proof: If you take a graph and find its vertex cover, the remaining vertices form an independent set,<br />
meaning there are no edges between any two vertices in the independent set, for if there were such an<br />
edge the rest of the vertices could not be a vertex cover.<br />
Clearly the smallest vertex cover gives the biggest independent set, and so the problems are equivallent -<br />
Delete the subset of vertices in one from the total set of vertices to get the order!<br />
Thus finding the maximum independent set must be NP-complete!<br />
Listen To Part 25-3<br />
In an independent set, there are no edges between two vertices. In a clique, there are always between two<br />
vertices. Thus if we complement a graph (have an edge iff there was no edge in the original graph), a<br />
clique becomes an independent set and an independent set becomes a Clique!<br />
file:///E|/LEC/LECTUR16/NODE21.HTM (4 of 7) [19/1/2003 1:35:28]
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