Explain the P vs NP problem in computational complexity theory. Why is it considered one of the most important open problems in computer science and mathematics? Describe the relationships between complexity classes (P, NP, NP-complete, NP-hard, PSPACE) and provide examples of problems in each category. Discuss the implications if P equals NP versus P not equaling NP. Explore approaches that have been tried to solve this problem and why they have failed. Consider practical consequences of the P vs NP question in fields like cryptography, optimization, and artificial intelligence.