Suppose you got two linear programs.
They are the same except that one has a shifted objective by a positive constant
(1) $$\min c^Tx$$
(2) $$\min c^Tx + d$$
For (2) there exists a 10-approximation, that means a algorithm that produces a solution $s_2$ at maximum ten times as bad as the optimal solution $s_2^*$:
$$ \frac{s_2}{10} \le s_2^* $$
The Question is what approximation ratio has (1) if you got a approximate solution $s_2$?
What i came up with
$$ s_2 - d= s_1 $$ $$ s_2^* - d = s_1^* $$
$$ \frac{s_1 + d}{10} \leq s_1^* + d $$
$$ \frac{s_1 - 9d}{10} \leq s_1^* $$
but this doesn't seem correct: if $d > s_1$ this is negative.
Thank you