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 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 class—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.