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 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.