What does it mean for a minimization algorithm to be a 2-approximation?

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.

The four techniques are greedy strategies, LP relaxation and rounding, the primal-dual schema, and local search.

Conditions: Applies to the design of approximation algorithms for NP-hard problems.

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

FPTAS and PTAS allow arbitrarily close approximation. APX allows constant-factor approximation.

Conditions: Comparing standard approximation classes for optimization problems.

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.