Improved LP-based Approximation Algorithms for Facility Location with Hard Capacities

We present LP-based approximation algorithms for the capacitated facility location problem (CFL), a long-standing problem with intriguing unsettled approximability in the 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 of $288$ due to An et al. in~2014. For CFL with cardinality facility cost (CFL-CFC), we present an LP-based $4$-approximation algorithm, which not only surpasses the long-standing ratio of~$5$ due to Levi et al. that ages up for decades since 2004 but also unties the long-time match to the best approximation for CFL that is obtained via local search in 2012. 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.