  We prove that weighted circuit satisfiability for monotone or antimonotone
circuits has no fixed-parameter tractable approximation algorithm with any
approximation ratio function $\rho$, unless $FPT\neq W[1]$. In particular, not
having such an fpt-approximation algorithm implies that these problems have no
polynomial-time approximation algorithms with ratio $\rho(OPT)$ for any
nontrivial function $\rho$.
