Ratio Bound

How does the approximate solution compare to the optimal solution?

An approximation algorithm has a ratio bound p(n) if for any input of size n, the cost C of the approximate solution is within a factor p(n) of the cost C* of the optimal solution:

$$1 \leq p(n) \leq \max(\frac{C}{C*}, \frac{C*}{C})$$

where C/C* is used for minimization problems, and C*/C for maximization problems.

Alternatively, an approximation algorithm has a relative error bound:

$$\frac{\mid C \;-\; C* \mid}{C*} \;\leq\; \epsilon(n)$$

where $\epsilon$(n) $\leq$ p(n) - 1.