# Is the max clique problem NP-complete?

## Is the max 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.

How that the 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. 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.

Does clique cover NP-complete?

Finding a minimum clique cover is NP-hard, and its decision version is NP-complete. It was one of Richard Karp’s original 21 problems shown NP-complete in his 1972 paper “Reducibility Among Combinatorial Problems”.

### What is an example of an NP-complete problem?

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.

Are NP problems solvable?

A problem is assigned to the NP (nondeterministic polynomial time) class if it is solvable in polynomial time by a nondeterministic Turing machine. A P-problem (whose solution time is bounded by a polynomial) is always also NP. It is much easier to show that a problem is NP than to show that it is NP-hard. …

Is 3 CLIQUE NP-complete?

The main idea is that the structure of 3-SAT is rich enough for the literals/clauses to be interpreted as (groups) of vertices. This then allows us to convert instances of 3-SAT to instances of graph theoretic problems. We will show that CLIQUE is an NP complete problem.

## Is 3 clique NP-complete?

How do you know if it is a NP problem?

A problem is NP if it is easy to verify a solution. It might not be easy to find such a solution, but given a candidate solution, we can easily verify whether it is in fact a solution or not. NP does not apply to algorithms, it applies to problems. A problem is NP if it is easy to verify a solution.

Is traveling salesman NP-complete?

Traveling Salesman Optimization(TSP-OPT) is a NP-hard problem and Traveling Salesman Search(TSP) is NP-complete. However, TSP-OPT can be reduced to TSP since if TSP can be solved in polynomial time, then so can TSP-OPT(1).

### What happens if P vs NP is solved?

If P equals NP, every NP problem would contain a hidden shortcut, allowing computers to quickly find perfect solutions to them. But if P does not equal NP, then no such shortcuts exist, and computers’ problem-solving powers will remain fundamentally and permanently limited.

Is every problem in NP?

Since NP-complete problems are in NP, their running time is at most exponential. However, some problems have been proven to require more time, for example Presburger arithmetic. Of some problems, it has even been proven that they can never be solved at all, for example the Halting problem.

Is vertex cover NP-complete?

Its decision version, the vertex cover problem, was one of Karp’s 21 NP-complete problems and is therefore a classical NP-complete problem in computational complexity theory.

## How is the maximal clique decision problem NP complete?

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.

Is the max clique problem a real world problem?

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.

Is there an algorithm for the maximum clique problem?

The maximum clique problem may be solved using as a subroutine an algorithm for the maximal clique listing problem, because the maximum clique must be included among all the maximal cliques. In the k -clique problem, the input is an undirected graph and a number k.

### Is the problem of finding the maximum clique intractable?

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.