# How do you prove clique problem is NP-complete?

## How do you prove clique problem is NP-complete?

The Clique Decision Problem belongs to NP-Hard – A problem L belongs to NP-Hard if every NP problem is reducible to L in polynomial time. Now, let the Clique Decision Problem by C. To prove that C is NP-Hard, we take an already known NP-Hard problem, say S, and reduce it to C for a particular instance.

### Is maximum clique problem NP-complete?

Theorem 20.2 Max-Clique is NP-Complete. We then put an edge between two nodes if the partial assignments are consistent. Notice that the maximum possible clique size is m because there are no edges between any two nodes that correspond to the same clause c.

Is K clique NP-complete?

Due to the way we constructed G, the k-clique must contain one vertex from each clause-group. For each of the vertices in the k-clique, select the corresponding literal in the corresponding clause of E. Thus our polynomial time reduction from CNF-SAT to k-Clique is answer preserving. Therefore k-Clique is NP-Complete.

How many steps are required to prove that a decision problem is NP-complete *?

Q. How many steps are required to prove that a decision problem is NP complete?
C. 3
D. 4
Explanation: first, the problem should be np. next, it should be proved that every problem in np is reducible to the problem in question in polynomial time.

## Is 5 clique NP-complete?

CLIQUE is NP-complete. Proof. 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.

### What is NP-complete with example?

NP-complete problem, any of a class of computational problems for which no efficient solution algorithm has been found. Many significant computer-science problems belong to this class—e.g., the traveling salesman problem, satisfiability problems, and graph-covering problems.

What is NP completeness explain?

(definition) Definition: The complexity class of decision problems for which answers can be checked for correctness, given a certificate, by an algorithm whose run time is polynomial in the size of the input (that is, it is NP) and no other NP problem is more than a polynomial factor harder.

Which of the following problems is NP-complete *?

Explanation: Hamiltonian circuit, bin packing, partition problems are NP complete problems. Halting problem is an undecidable problem.

## What is meant by NP complete problem?

NP-complete problem, any of a class of computational problems for which no efficient solution algorithm has been found. If a problem is NP and all other NP problems are polynomial-time reducible to it, the problem is NP-complete.

### 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 ).

How do you prove that a problem is NP complete?

Correspondingly, the Clique Decision Problem is to find if a clique of size k exists in the given graph or not. To prove that a problem is NP-Complete, we have to show that it belongs to both NP and NP-Hard Classes. (Since NP-Complete problems are NP-Hard problems which also belong to NP)

## What is a clique decision problem?

The Clique Decision Problem belongs to NP – If a problem belongs to the NP class, then it should have polynomial-time verifiability, that is given a certificate, we should be able to verify in polynomial time if it is a solution to the problem. Certificate – Let the certificate be a set S consisting of nodes in the clique and S is a subgraph of G.