Prove that the polynomial time reduction is from the Clique problem to the Vertex Cover problem

Data Structure AlgorithmsComputer ScienceComputers

Vertex cover is a subset of vertices that covers all the edges in a graph. It is used to determine whether a given graph has a 3SAT to vertex cover.

Clique is called a subset of vertices that are all directly connected. It determines whether a clique of size k exists in a graph.

To prove − Vertex cover can be reduced to clique.

Proof

Given a graph G=(V,E) and integer k.

Get its complement graph G'=(V,E').

Solve CLIQUE(G',|V|-k).

If there is a solution, return yes. Otherwise, it returns as no.

To prove this reduction, we need to show the following −

  • If there is a solution to VERTEX-COVER(G,k), then there must be a solution to CLIQUE(G',|V|-k) and vice versa.

  • Assume that G has a vertex cover V' ⊆ V , where |V |' = k. Then, for all u, v ε V , if (u, v) ε E, then u ε V' or v ε V' or both since the vertex cover must cover all edges.

  • The contrapositive is that for all u, v ε V, if u does not belong to V', then (u, v) ε/ E and thus (u, v) ε E.

  • For any pair of vertices that are both not in the vertex cover V' of G, there is an edge between them in G.

  • The union of all pairs of vertices that are all not in V' is simply V − V'. Thus, V − V' is a clique in G, by definition, and V − V' has size |V | − k.

  • This operation can be done in polynomial time. Since VERTEX-COVER can be reduced to CLIQUE in polynomial time, CLIQUE ε NP and VERTEX-COVER is NP-complete, CLIQUE is also NP-Complete.

raja
Published on 15-Jun-2021 12:27:06
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