Computational Theory Questions
The main difference between NP-complete and NP-hard problems lies in their level of difficulty.
An NP-complete problem is a decision problem that belongs to the class of problems known as NP (nondeterministic polynomial time). These problems are considered to be the most difficult problems in NP, as they have the property that any other problem in NP can be reduced to them in polynomial time. In other words, if a polynomial-time algorithm exists for solving an NP-complete problem, then it can be used to solve any other problem in NP efficiently.
On the other hand, an NP-hard problem is a broader category that includes both NP-complete problems and problems that are even more difficult. NP-hard problems do not necessarily have to be decision problems, and they may not have a known polynomial-time algorithm for solving them. While NP-complete problems have the property of being reducible to each other, NP-hard problems do not necessarily have this property.
In summary, all NP-complete problems are NP-hard, but not all NP-hard problems are NP-complete. NP-complete problems are the most difficult problems in NP, while NP-hard problems encompass a wider range of difficult problems.