Let $X$ and $Y$ be two NPO problems. Let $(f,g)$ be a reduction between $X$ and $Y$,in particular, assume that $(f,g)$ is both P-reduction and A-reduction, i.e., there exist two poly-time  computable functions $c_1,c_2 :(1, +\infty)\cap \mathbb{Q}\mapsto (1, +\infty)\cap \mathbb{Q}$ such that:$R_B(f(x),y)\leq r \Rightarrow R_A(x, g(x,y))\leq c_2(r)$ (A-reduction) and$R_B(f(x),y)\leq c_1(r) \Rightarrow R_A(x, g(x,y))\leq r$ (P-reduction) Then, can you prove that $c_1$ is invertible and that $c_1=c_2^{-1}$ is always the case ?(I am reading a paper which asserts that this is always the case, but I don't see it true...) 
On approx-preserving P- and A-reducibilities
ds.algorithms;np hardness;approximation algorithms;approximation hardness