Start by recalling the definitions of P, NP, NP-hard, and NP-complete classes, and understand how they relate to each other in the complexity hierarchy.

Compare the Traveling Salesman Problem (TSP) and Boolean Satisfiability Problem (SAT) to illustrate the difference between NP-hard and NP-complete problems, considering their decision and optimization variants.

Prove why the Halting Problem is undecidable and explain how this conceptually differs from NP-hardness, then discuss the implications of NP-hardness on algorithm design and computational feasibility.