What is a clique independent set?

What is a clique independent set?

1 Cliques and Independent Sets. Definition 1. A set of vertices is called independent if no two vertices in the set are adjacent. A set of vertices is called a clique if every two vertices in the set are adjacent.

Is independent set NP-Complete?

The independent set decision problem is NP-complete, and hence it is not believed that there is an efficient algorithm for solving it. The maximum independent set problem is NP-hard and it is also hard to approximate.

Is clique opposite of independent set?

clique and independent set are not directly complementary graphs of eachother, but the same set of vertices mean two different things based on the complement of the bigger graph..

What is a clique NP-Complete?

Prerequisite: NP-Completeness. A clique is a subgraph of a graph such that all the vertices in this subgraph are connected with each other that is the subgraph is a complete graph.

What is clique in algorithm?

By convention, in algorithm analysis, the number of vertices in the graph is denoted by n and the number of edges is denoted by m. A clique in a graph G is a complete subgraph of G. That is, it is a subset K of the vertices such that every two vertices in K are the two endpoints of an edge in G.

Is a complete graph a clique?

A complete graph is often called a clique. The size of the largest clique that can be made up of edges and vertices of G is called the clique number of G.

Is clique 3 NP-complete?

a) Prove that CLIQUE-3 is in NP. Verifying each node is in polynomial time since there are a maximum number of edges and nodes in a clique. There also can only be a polynomial number of cliques in a graph due to the limit of 3 edges on each vertex. Therefore, the verifier runs in polynomial time and CLIQUE-3 is in NP.

Is clique problem NP-complete?

The clique decision problem is NP-complete (one of Karp’s 21 NP-complete problems). The problem of finding the maximum clique is both fixed-parameter intractable and hard to approximate.

Is clique decision problem NP-complete?

The clique decision problem is NP-complete (one of Karp’s 21 NP-complete problems). The problem of finding the maximum clique is both fixed-parameter intractable and hard to approximate. And, listing all maximal cliques may require exponential time as there exist graphs with exponentially many maximal cliques.

What is clique decision?

In the field of computer science, the clique decision problem is a kind of computation problem for finding the cliques or the subsets of the vertices which when all of them are adjacent to each other are also called complete subgraphs.

Are the clique and independent set problems NP-complete?

We know that both the clique and independent set problems are NP-Complete in of themselves. We also know that the verification of this problem, given some “certificate” is in NP.

Is the independent set NP-complete or NP-hard?

Therefore, any instance of the independent set problem can be reduced to an instance of the clique problem. Thus, the independent set is NP-Hard. Since the Independent Set problem is both NP and NP-Hard, therefore it is an NP-Complete problem.

How to prove that the clique problem is NP-hard?

To prove that the clique problem is NP-Hard, we take the help of a problem that is already NP-Hard and show that this problem can be reduced to the Clique problem. For this, we consider the Independent Set problem, which is NP-Complete (and hence NP-Hard ).

What is an instance of an independent set problem?

An instance of the problem is an input specified to the problem. An instance of the Independent Set problem is a graph G (V, E) and a positive integer k, and the problem is to check whether an independent set of size k exists in G.