Solving Decision Problem with k Colors - NP-Complete

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In summary, the conversation discusses a test question about a decision problem for coloring with at most $k$ colors and its NP-completeness. The decision problem is defined as finding a coloring with $k$ colors and it is shown to be in NP by using a non-deterministic Turing machine. The completeness part is missing, but it is later clarified that the exercise only requires an explanation, not a proof, for the NP-completeness of 3-coloring. The conversation also mentions the possibility of reducing 3SAT to 3-coloring to prove its NP-completeness.
  • #1
evinda
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Hello! (Wave)

I had a test today and a question was to make a decision problem for the coloring using at most $k$ colors and to show that this problem is NP-complete.

Is the following right?

Decision problem: Is there a coloring with $k$ colors?

Also could we show that the problem is in NP as follows?

A non-deterministic Turning machine first guesses the nodes that are colored with each of the $k$ colors and then it verifies in $\Theta(E)$ time that each adjacent nodes of the graph have a different colour.
Thus the problem is in NP.
 
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  • #2
This seems OK. There is still the completeness part missing.
 
  • #3
Evgeny.Makarov said:
This seems OK.

(Happy)

Evgeny.Makarov said:
There is still the completeness part missing.

Oh, I am sorry.. we had just to prove that it is in NP, not that it is NP-complete.Then the exercise required to explain, but not to prove if 3-coloring is NP-complete.

I wrote that it is NP-complete, because it can be reduced to the 3SAT problem, which is known to be NP-complete. Do you think that it is enough? (Thinking)
 
  • #4
evinda said:
Then the exercise required to explain, but not to prove if 3-coloring is NP-complete.

I wrote that it is NP-complete, because it can be reduced to the 3SAT problem, which is known to be NP-complete.
Reducing 3-coloring to 3SAT does not help in any way.
 
  • #5
Evgeny.Makarov said:
Reducing 3-coloring to 3SAT does not help in any way.

Oh yes, right... (Tmi)
We could say that we can reduce 3SAT to 3-coloring, right?
 
  • #6
Yes. I don't know the details of the reduction, though.
 
  • #7
Evgeny.Makarov said:
Yes. I don't know the details of the reduction, though.

Ok, no problem... (Smile)

Thanks a lot for your answer! (Happy)
 

1. What is NP-Complete?

NP-Complete stands for Non-deterministic Polynomial-Complete, which is a complexity class in computer science used to classify problems that are considered difficult to solve. NP-Complete problems are those that require a large amount of time to solve, and no known efficient algorithm exists for solving them.

2. How does the "Solving Decision Problem with k Colors" relate to NP-Complete?

The "Solving Decision Problem with k Colors" is a specific problem that falls under the NP-Complete complexity class. It involves determining if a given graph can be colored using k colors, without any adjacent vertices having the same color. This problem is known to be NP-Complete, meaning that it is difficult to solve and no known efficient algorithm exists for solving it.

3. What is the significance of solving the "Solving Decision Problem with k Colors" in the field of computer science?

Solving the "Solving Decision Problem with k Colors" problem is significant in the field of computer science because it is an NP-Complete problem. Finding an efficient algorithm for solving this problem would not only solve this specific problem, but it would also lead to advancements in other NP-Complete problems, which are commonly encountered in real-world scenarios.

4. What are some approaches for solving the "Solving Decision Problem with k Colors"?

There are several approaches for solving the "Solving Decision Problem with k Colors". One approach is using a brute-force algorithm, which involves trying all possible color combinations until a valid coloring is found. Another approach is using a heuristic algorithm, which involves using a set of rules or guidelines to guide the coloring process. Other approaches include using mathematical formulations and optimization techniques.

5. Can the "Solving Decision Problem with k Colors" be solved in polynomial time?

As an NP-Complete problem, it is currently believed that the "Solving Decision Problem with k Colors" cannot be solved in polynomial time. This means that the time required to solve the problem increases exponentially as the size of the graph increases. However, researchers are constantly working on developing more efficient algorithms and techniques for solving NP-Complete problems, so this may change in the future.

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