Nowadays,　two　issues　in　data　transmission　for　networks　are　attracting　considerable　interest　in　the　research　community:　energy　efficiency　and　cost　minimization,　i.　e.,　to　minimize　the　energy　consumed　and　resource　occupied.　This　paper　considers　approximation　algorithms　for　the　energy　constrained　minimum　cost　Steiner　tree　(ECMST)　problem　which　have　applications　in　energy-efficient　minimum　cost　multicast.　Let　G　=　(V,　E)　be　a　given　undirected　graph,　S　⊆　V　be　a　terminal　set,　and　c:　E　→　Z+0　and　d:　E　→　Z+0　respectively　be　the　cost　function　and　energy　consumption　function　for　the　edges.　For　a　threshold　D,　ECMST　is　to　compute　a　minimum　cost　tree　spanning　all　specified　terminals　of　S,　with　its　total　energy　consumption　bounded　by　D.　This　paper　first　shows　that　ECMST　is　pseudo-polynomial　solvable　when　the　number　of　the　terminals　are　fixed.　Then　it　presents　a　polynomial　time　factor-　(2(1+　1/k),　2(1+k))　approximation　algorithm　via　Lagrangian　Relaxation　for　any　k　＞　0.　Last　but　not　the　least,　by　a　more　sophisticated　application　of　Lagrangian　relaxation　technique,　we　obtain　an　approximation　algorithm　with　ratio　(2,　2　+　Ε)　for　any　fixed　Ε　＞　0.　©　Springer　International　Publishing　Switzerland　2015.