Approximation Algorithms provide feasible solutions with guaranteed worst-case performance bounds for NP-hard optimization problems. An α-approximation algorithm for minimization ensures: solution value ≤ α × optimal value. For maximization: solution ≥ (1/α) × optimal. Examples: (1) Christofides algorithm for metric TSP (1.5-approximation); (2) greedy set cover (ln-approximation); (3) vertex cover (2-approximation). Performance ratio analysis balances: solution quality, computational complexity, and practical effectiveness. Some problems admit polynomial-time approximation schemes (PTAS) - algorithms achieving (1+ε)-approximation for any ε > 0. Others are APX-hard, meaning constant-factor approximation is best possible (unless P=NP).