Given an algorithm with approximation ratio $\alpha$, and another algorithm with approximation ratio $\beta=n^\epsilon$, and a solution to a problem with cost $c$. What is the standard way to bound $\alpha$ in terms of $\beta$ if$(n-c) \leq \alpha (n- \beta c)$. That is, what can I say about $\alpha$ in terms of $\beta$ (assuming all positives, and $c < n$).
Bounding the cost of an approximation algorithm when subtraction involve
np hardness;approximation;analysis of algorithms