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An approximation bound is a guaranteed multiplicative limit on how much worse the algorithm's result can be than the true optimum in the worst case.

Conditions: Used for minimization or maximization problems.; Expresses a worst-case guarantee relative to the optimum.

An approximation bound is a guaranteed multiplicative limit on how much worse the algorithm's result can be than the true optimum in the worst case.

Conditions: Used for minimization or maximization problems.; Expresses a worst-case guarantee relative to the optimum.

For minimization problems, the guarantee is $C \leq \rho(n) \cdot C^*$. For maximization problems, it is $C \geq \frac{C^*}{\rho(n)}$.

Conditions: C is the algorithm-produced cost.; C* is the true optimal cost.; rho(n) is the approximation bound.

For minimization problems, the guarantee is $C \leq \rho(n) \cdot C^*$. For maximization problems, it is $C \geq \frac{C^*}{\rho(n)}$.

Conditions: C is the algorithm-produced cost.; C* is the true optimal cost.; rho(n) is the approximation bound.

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

The algorithm selects a set of disjoint edges M. The optimal cover must include at least one vertex per edge in M, so optimal cost >= |M|.

Conditions: The algorithm selects arbitrary edges and adds both endpoints until no edges remain

The algorithm selects a set of disjoint edges M. The optimal cover must include at least one vertex per edge in M, so optimal cost >= |M|.

Conditions: The algorithm selects arbitrary edges and adds both endpoints until no edges remain

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