Provide a comprehensive exploration of the P vs NP problem, starting with formal definitions of P, NP, NP-complete, and NP-hard classes. Explain the significance of this problem for computer science and mathematics. Discuss the concept of polynomial time reductions and their role in defining NP-completeness. Present classic examples of NP-complete problems including SAT, 3SAT, Hamiltonian Path, and Graph Coloring. Explore alternative approaches to the P vs NP question including circuit complexity, proof complexity, and relativization barriers. Discuss the implications of both possible resolutions of this problem for cryptography, optimization, and our understanding of computation.