The same books on theory of computation I recommended on other occasions.mathmari said:
Evgeny.Makarov said:
NP-Complete problems are a class of computational problems that are believed to be among the most difficult to solve. They are part of the larger class of NP (nondeterministic polynomial time) problems, which refers to problems that can be verified in polynomial time but may not necessarily be solved in polynomial time.
One way to recognize if a problem is NP-Complete is by using the concept of reduction. This involves reducing a known NP-Complete problem to the problem in question, and if the reduction can be done in polynomial time, then the problem is also considered NP-Complete. Another way is to check if the problem is both NP and NP-Hard, which essentially means it is at least as difficult as other NP problems.
Yes, there are many real-world applications for NP-Complete problems. Some examples include scheduling and optimization problems, network routing and design, and resource allocation. NP-Complete problems are also commonly used in cryptography and security systems.
Currently, there is no known algorithm that can efficiently solve all NP-Complete problems. However, some NP-Complete problems may have efficient algorithms for certain special cases or approximations. Researchers are constantly working on developing faster and more efficient algorithms for solving these difficult problems.
Understanding NP-Complete problems can help in solving other computational problems by providing insights into the complexity of different problems and the limitations of current algorithms. It can also help in identifying similarities between different problems and finding efficient ways to solve them. Additionally, understanding NP-Complete problems is important for developing new algorithms and advancing the field of computer science.