Discuss the implications of NP-completeness for the time complexity of solving optimization problems, using the Traveling Salesman Problem (TSP) as a case study.

NP-completeness implies that no known polynomial-time algorithm can solve the problem efficiently for all instances, as with the TSP. If a polynomial-time algorithm is found for any NP-complete problem, it would apply to all NP problems, revolutionizing computational complexity theory. Currently, TSP solutions often rely on heuristics or approximation algorithms due to their impracticality for large inputs with exact algorithms.