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Finding an exact solution for NP-hard optimization problems requires exponential time, such as $O(2^n)$ or $O(n!)$, which is computationally infeasible for large inputs.

Conditions: Problem is an NP-hard optimization problem; Seeking an exact solution

Finding an exact solution for NP-hard optimization problems requires exponential time, such as $O(2^n)$ or $O(n!)$, which is computationally infeasible for large inputs.

Conditions: Problem is an NP-hard optimization problem; Seeking an exact solution

An approximation algorithm is a polynomial-time algorithm designed to find near-optimal solutions for NP-hard optimization problems. It trades absolute perfection for mathematical guarantees, ensuring provable efficiency.

Conditions: Applies to NP-hard optimization problems; Runs in polynomial time

An approximation algorithm is a polynomial-time algorithm designed to find near-optimal solutions for NP-hard optimization problems. It trades absolute perfection for mathematical guarantees, ensuring provable efficiency.

Conditions: Applies to NP-hard optimization problems; Runs in polynomial time