Is clique in 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 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.
How do you show clique in NP?
To show CLIQUE is in NP, our verifier takes a graph G(V,E), k, and a set S and checks if |S| ≥ k then checks whether (u, v) ∈ E for every u, v ∈ S. Thus the verification is done in O(n2) time.
Is clique a complete graph?
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.
What is clique size?
A clique of size is called a -clique (though this term is also sometimes used to mean a maximal set of vertices that are at a distance no greater than. from each other). 0-cliques correspond to the empty set (sets of 0 vertices), 1-cliques correspond to vertices, 2-cliques to edges, and 3-cliques to 3-cycles.
How do you verify a clique?
Verification – We have to check if there exists a clique of size k in the graph. Hence, verifying if number of nodes in S equals k, takes O(1) time. Verifying whether each vertex has an out-degree of (k-1) takes O(k2) time. (Since in a complete graph, each vertex is connected to every other vertex through an edge.
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.
Which of the following problems is not NP-complete?
Which of the following problems is not NP complete? Explanation: Hamiltonian circuit, bin packing, partition problems are NP complete problems. Halting problem is an undecidable problem.
What is clique in algorithm?
Algorithms clique A clique is a subset of vertices of an undirected graph G such that every two distinct vertices in the clique are adjacent; that is, its induced subgraph is complete. Cliques are one of the basic concepts of graph theory and are used in many other mathematical problems and constructions on graphs.
Is a clique fully connected?
Recalling that a clique is a subset C of vertices of an undirected graph such that the subgraph induced by C is fully connected. That is, every two distinct vertices in C are connected by a distinct edge of the graph.
How do I know my clique size?
To find a clique of G:
- Suppose that G has n vertices.
- Find a vertex v of the smallest possible degree in G.
- If the degree of v is n − 1, stop; G is a clique, so the largest clique in G has size n.
- Otherwise, remove v and all of its edges from G. Find the largest clique in the smaller graph.
What is NP-hard but not NP-complete?
There are decision problems that are NP-hard but not NP-complete such as the halting problem. That is the problem which asks “given a program and its input, will it run forever?” That is a yes/no question and so is a decision problem. It is easy to prove that the halting problem is NP-hard but not NP-complete.
Which of the following is not Undecidable NP-complete?
Detailed Solution The halting problem is NP-Hard, not NP-Complete, but is undecidable. Hence option 2 is correct. Hamiltonian circuit, bin packing, partition problems are NP-complete problems.
What is the maximum size of a clique?
The “maximum size clique” for a graph of n vertices is a clique of the largest size k (k ≤ n) such that there does not exist a clique of size k + 1 in the graph. A “maximal size clique for a vertex i” in a graph is the clique of the largest size that involves vertex i as one of the constituent vertices.
Is clique detection problem is NP-complete?
In this article, we will prove that the Clique Detection Problem is NP-Complete by the help of Independent Set problem, which is NP-Complete. Refer to Proof that Clique Decision problem is NP-Complete, for the proof with the help of Boolean Satisfiability 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 the clique problem?
An instance of the problem is an input specified to the problem. An instance of the Clique problem is a graph G (V, E) and a positive integer K, and the problem is to check whether a clique of size K exists in G.
What is the difference between maximal clique problem and clique decision problem?
The Maximal Clique Problem is to find the maximum sized clique of a given graph G, that is a complete graph which is a subgraph of G and contains the maximum number of vertices. This is an optimization problem. Correspondingly, the Clique Decision Problem is to find if a clique of size k exists in the given graph or not.