# Is maximum 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. And, listing all maximal cliques may require exponential time as there exist graphs with exponentially many maximal cliques.

Table of Contents

## Is maximum 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. And, listing all maximal cliques may require exponential time as there exist graphs with exponentially many maximal cliques.

## Is Max clique NP-hard?

MaxClique is NP-Hard. Proof : We show a reduction from 3SAT. So, consider an input to 3SAT, which is a formula F defined over n variables (and with m clauses).

**Why is the clique problem NP-complete?**

The Boolean Satisfiability Problem (S) is an NP-Complete problem as proved by the Cook’s theorem. Therefore, every problem in NP can be reduced to S in polynomial time. Thus, if S is reducible to C in polynomial time, every NP problem can be reduced to C in polynomial time, thereby proving C to be NP-Hard.

### What is the meaning of NP-complete?

A problem is called NP (nondeterministic polynomial) if its solution can be guessed and verified in polynomial time; nondeterministic means that no particular rule is followed to make the guess. If a problem is NP and all other NP problems are polynomial-time reducible to it, the problem is NP-complete.

### What is the difference between NP-hard and NP-complete?

A Problem X is NP-Hard if there is an NP-Complete problem Y, such that Y is reducible to X in polynomial time….Difference between NP-Hard and NP-Complete:

NP-hard | NP-Complete |
---|---|

To solve this problem, it do not have to be in NP . | To solve this problem, it must be both NP and NP-hard problems. |

**Are NP-Hard problems NP-complete?**

A problem X is NP-Complete if there is an NP problem Y, such that Y is reducible to X in polynomial time. NP-Complete problems are as hard as NP problems….Difference between NP-Hard and NP-Complete:

NP-hard | NP-Complete |
---|---|

To solve this problem, it do not have to be in NP . | To solve this problem, it must be both NP and NP-hard problems. |

#### Which problems are NP-complete?

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 classâ€”e.g., the traveling salesman problem, satisfiability problems, and graph-covering problems.

#### How many cliques are in a complete graph?

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. , etc….Clique.

graph family | OEIS | number of cliques |
---|---|---|

complete bipartite graph | A000290 | 4, 9, 16, 25, 36, 49, 64, 81, 100. |

**What is a maximum clique in a graph?**

A maximal clique is a clique that cannot be extended by including one more adjacent vertex, meaning it is not a subset of a larger clique. A maximum clique (i.e., clique of largest size in a given graph) is therefore always maximal, but the converse does not hold.

## What is wrong with the following proof of NP completeness for clique 3?

b) What is wrong with the following proof of NP-completeness for CLIQUE-3? This isn’t actually proving anything about CLIQUE-3, it’s doing the reduction incorrectly. What this is saying is that CLIQUE is at least as hard as CLIQUE-3, not the other way around.

## What is NP-complete example?

**Is clique decision problem NP-complete or NP-hard?**

Hence, for a particular instance, the satisfiability problem is reduced to the clique decision problem. Therefore, the Clique Decision Problem is NP-Hard. The Clique Decision Problem is NP and NP-Hard. Therefore, the Clique decision problem is NP-Complete.

### 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.

### What is the difference between complete sub-graph and max clique?

Complete sub-graph means, all the vertices of this sub-graph is connected to all other vertices of this sub-graph. The Max-Clique problem is the computational problem of finding maximum clique of the graph. Max clique is used in many real-world problems.

**What is a max clique in math?**

DAA – Max Cliques. In an undirected graph, a clique is a complete sub-graph of the given graph. Complete sub-graph means, all the vertices of this sub-graph is connected to all other vertices of this sub-graph.