  We present LP-based approximation algorithms for the capacitated facility
location problem (CFL), a long-standing problem with intriguing unsettled
complexity and literature dated back to the 90s. We present an elegant
iterative rounding scheme for the MFN relaxation that yields an approximation
guarantee of $\left(10+\sqrt{67}\right)/2 \approx 9.0927$, a significant
improvement upon the previous LP-based ratio due to An et al in~2014. For CFL
with cardinality facility cost (CFL-CFC), we present an LP-based
$4$-approximation algorithm, which surpasses the long-standing ratio of~$5$ due
to Levi et al that ages up for decades since 2004. Our result considerably
deepens the current understanding for the CFL problem and indicates that an
LP-based ratio strictly better than $5$ in polynomial time for the general
problem may still be possible to pursue.
