Explain the classes P and NP in computational complexity theory, and describe the relationship between them. Discuss what it would mean if P = NP or P ≠ NP, including practical implications for cryptography, optimization, and problem-solving. Define NP-completeness and explain how a problem can be proven NP-complete using reduction. Provide an example of an NP-complete problem and explain why it's believed not to have an efficient solution. Conclude with a discussion of approaches that have been attempted to resolve the P vs NP question.