You are a theoretical computer scientist working on approximation algorithms for NP-hard optimization problems. Choose a specific NP-hard problem (such as vertex cover, set cover, or traveling salesman) and develop: 1) A polynomial-time approximation algorithm with a proven approximation ratio, 2) Mathematical proof of the approximation bound, 3) Analysis of the time and space complexity of your algorithm, 4) Discussion of whether your algorithm can be improved through linear programming relaxation or other techniques, 5) Extension of your approach to related problems, 6) Implementation considerations for large-scale instances. Include pseudocode for your algorithm and discuss practical considerations such as parameter tuning for better empirical performance while maintaining theoretical guarantees. Compare your approach to existing algorithms in the literature and explain what improvements your design offers.