Approximation algorithms for the fault-tolerant facility location problem with submodular penalties

This work is to discuss the fault-tolerant facility location problem with submodular penalties. We propose an LP-rounding 2.27-approximation algorithm, where every demand point j has a requirement that \(t_{j}\) $t_j$ distinct facilities serve it. This is the first constant performance guarantee known for this problem. In addition, we give an LP-rounding 2-approximation algorithm for the case where all requirements are the same.