On Facility Location with General Lower Bounds

In this paper, we give the first constant approximation algorithm for the lower bounded facility location (LBFL) problem with general lower bounds. Prior to our work, such algorithms were only known for the special case where all facilities have the same lower bound: Svitkina \cite{Svi10} gave a $448$-approximation for the special case, and subsequently Ahmadian and Swamy \cite{AS13} improved the approximation factor to 82.6. As in \cite{Svi10} and \cite{AS13}, our algorithm for LBFL with general lower bounds works by reducing the problem to the capacitated facility location (CFL) problem. To handle some challenges caused by the general lower bounds, our algorithm involves more reduction steps. One main complication is that after aggregation of clients and facilities at a few locations, each of these locations may contain many facilities with different opening costs and lower bounds. To handle this issue, we introduce and reduce our LBFL problem to an intermediate problem called the transportation with configurable supplies and demands (TCSD) problem, which in turn can be reduced to the CFL problem.