Can we reduce the problem of clique to the problem of 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.

Is the complement of vertex cover?

Your decision problem, NO-VERTEX-COVER , is exactly the complement of the well-known VERTEX-COVER problem; the latter problem is NP -complete (and is, of course, in NP ). Your problem NO-VERTEX-COVER is thus in co-NP . (Recall that a problem is in co-NP if and only if its complement is in NP .)

What is vertex cover problem find a vertex cover of minimum size in a given undirected graph?

In an undirected graph, G = (V, E), where V is the set of vertices and E is the set of edges. A vertex cover (VC) is a subset S ⊆ V in which each edge in G has at least one endpoint in S. The minimum vertex cover (MVC) problem is to find the minimum size of the vertex cover in a graph.

Can Set Cover be reduced to vertex cover?

(The instance of the Set Cover problem that we derive from the Vertex Cover instance, is known as a reduced instance. Hence, the answer to the Vertex Cover instance is also “yes”. In other words: If the answer to the reduced Set Cover instance is “yes”, then the answer to the original Vertex Cover instance is “yes”.

How do you reduce independent set to vertex cover?

The reduction from vertex cover to independent set is function f given by f(G,k) = (G,n-k) where n is the number of vertices in G. Function f is clearly computable in polynomial time.

What is vertex cover set?

In graph theory, a vertex cover (sometimes node cover) of a graph is a set of vertices that includes at least one endpoint of every edge of the graph.

What is vertex cover problem give an approximation algorithm for vertex cover problem?

An optimal vertex cover is {b, c, e, i, g}. As it turns out, this is the best approximation algorithm known for vertex cover. It is an open problem to either do better or prove that this is a lower bound. Observation: The set of edges picked by this algorithm is a matching, no 2 edges touch each other (edges disjoint).

What is vertex cover problem prove that vertex cover problem is NP complete?

Proof that vertex cover is in NP – If any problem is in NP, then, given a ‘certificate’ (a solution) to the problem and an instance of the problem (a graph G and a positive integer k, in this case), we will be able to verify (check whether the solution given is correct or not) the certificate in polynomial time.

Is Set Cover NP-hard?

The decision version of set covering is NP-complete, and the optimization/search version of set cover is NP-hard. If each set is assigned a cost, it becomes a weighted set cover problem.

How do you reduce independent sets to cliques?

By definition of the complement graph, u, v is connected in G iff u, v is not connected in G. Hence, S is an independent set in G iff S is a clique in G . Hence, if the answer to INDEPENDENT SET is YES iff the answer to CLIQUE is YES. Hence, INDEPENDENT SET reduces to CLIQUE.