We consider the facility location problem with submodular penalties (FLPSP) and the facility location problem with linear penalties (FLPLP), two extensions of the classical facility location problem (FLP). First, we introduce a general algorithmic framework for a class of covering problems with submodular penalties, extending the recent result of Geunes et al. (Math Program 130:85–106, 2011) with linear penalties. This framework leverages existing approximation results for the original covering problems to obtain corresponding results for their counterparts with submodular penalties. Specifically, any LP-based α\documentclass[12pt]{minimal}
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\begin{document}$$\alpha $$\end{document}-approximation for the original covering problem can be leveraged to obtain an 1-e-1/α-1\documentclass[12pt]{minimal}
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\begin{document}$$\left( 1-e^{-1/\alpha }\right) ^{-1}$$\end{document}-approximation algorithm for the counterpart with submodular penalties. Consequently, any LP-based approximation algorithm for the classical FLP (as a covering problem) can yield, via this framework, an approximation algorithm for the counterpart with submodular penalties. Second, by exploiting some special properties of submodular/linear penalty function, we present an LP rounding algorithm which has the currently best 2\documentclass[12pt]{minimal}
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\begin{document}$$2$$\end{document}-approximation ratio over the previously best 2.375\documentclass[12pt]{minimal}
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\begin{document}$$2.375$$\end{document} by Li et al. (Theoret Comput Sci 476:109–117, 2013) for the FLPSP and another LP-rounding algorithm which has the currently best 1.5148\documentclass[12pt]{minimal}
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\begin{document}$$1.5148$$\end{document}-approximation ratio over the previously best 1.853\documentclass[12pt]{minimal}
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\begin{document}$$1.853$$\end{document} by Xu and Xu (J Comb Optim 17:424–436, 2008) for the FLPLP, respectively.
