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 tj\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t_{j}$$\end{document} 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.