Is NP-Complete Harder to Solve than NP-Hard?

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In summary, the class of problems NP-Complete is more difficult to solve than the class of problems NP-Hard. NP-Complete problems are in NP and NP-Hard, while NP-Hard problems are not necessarily in NP. NP-Hard problems require more computational resources and may be impossible to solve in general. The class NP includes all problems that are easier than its "toughest" problems, making it easier to distinguish between NP-Complete and NP-Hard problems. This type of question is better suited for the Programming forum as it pertains to computer science.
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jetoso
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I have a question. Is the class of problems NP-Complete more difficult to solve than the class of problems NP-Hard?

I mean, NP-Complete problems are in NP, and also are NP-Hard, but not all NP-Hard problems are in NP... How to tell which one is more difficult to solve?
 
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The set of problems that are NP-hard is a strict superset of the problems that are NP-complete. Some problems (like the halting problem) are NP-hard, but not in NP.
 
  • #3
NP-Hard problems that are not NP-Complete are 'harder' than NP problems, in general, in that they take up more computational resources of some type: memory, time, alternation, randomness, etc.
 
  • #4
Or as Nate mentioned, NP-hard problems might be impossible to solve in general. You can tell the difference in difficulty between NP-hard and NP-complete problems because the class NP includes everything easier than its "toughest" problems--if a problem is not in NP, it is harder than all the problems in NP.

By the way, would you post things like this in the future in the Programming forum, as it is a computer science question?
 
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Related to Is NP-Complete Harder to Solve than NP-Hard?

1. What is the difference between NP-hard and NP-Complete?

NP-hard and NP-Complete are both complexity classes in computer science, but NP-Complete is a subset of NP-hard. NP-hard problems are those that are at least as hard as the hardest problems in the class NP, while NP-Complete problems are those that are both NP-hard and also in the class NP.

2. How do you determine if a problem is NP-hard or NP-Complete?

To determine if a problem is NP-hard or NP-Complete, you need to analyze its time complexity. If the problem can be solved in polynomial time, it is not considered NP-hard or NP-Complete. If the problem can be reduced to another NP-hard problem, it is considered NP-hard. If the problem can also be verified in polynomial time, it is considered NP-Complete.

3. What makes NP-hard and NP-Complete problems difficult to solve?

NP-hard and NP-Complete problems are difficult to solve because they require a large amount of time and computational resources to find a solution. In fact, these problems are considered to be among the most challenging problems in computer science, as there is no known efficient algorithm that can solve them.

4. Are there any real-world applications for NP-hard and NP-Complete problems?

Yes, there are many real-world applications for NP-hard and NP-Complete problems. Some examples include scheduling problems, logistics and routing problems, and optimization problems in various industries such as transportation, manufacturing, and finance. These problems are also used in cryptography and game theory.

5. Is there ongoing research in the field of NP-hard and NP-Complete problems?

Yes, there is ongoing research in the field of NP-hard and NP-Complete problems. As these problems are difficult to solve and have many real-world applications, there is a constant effort to develop more efficient algorithms and techniques for solving them. This research is interdisciplinary and involves computer scientists, mathematicians, and engineers.

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