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　-approximation　for　the　original　covering　problem　can　be　leveraged　to　obtain　an　-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　-approximation　ratio　over　the　previously　best　by　Li　et　al.　(Theoret　Comput　Sci　476:109-117,　2013)　for　the　FLPSP　and　another　LP-rounding　algorithm　which　has　the　currently　best　-approximation　ratio　over　the　previously　best　by　Xu　and　Xu　(J　Comb　Optim　17:424-436,　2008)　for　the　FLPLP,　respectively.