Clouds,　such　as　Amazon　Infrastructure-as-a-Service　(IaaS)　clouds　and　EMC　Hybrid　Cloud,　impose　growing　requirements　of　resource-efficiency　scheduling.　The　bounded　flexible　scheduling　(BFS)　problem　is　one　of　the　problems　proposed　to　meet　such　requirements.　In　BFS,　we　are　given　a　set　of　identical　machines　and　a　set　of　jobs,　each　of　which　is　with　a　value,　a　workload,　a　deadline　and　a　parallelism　degree,　i.e.,　the　maximum　number　of　machines　on　which　the　job　can　execute　concurrently.　The　problem　is　to　compute　an　assignment　of　the　given　jobs　to　the　machines,　such　that　the　total　value　of　the　jobs　successfully　completed　by　their　deadlines　is　maximized.　This　paper　presents　a　factor-C-k/C　approximation　algorithm　for　BFS,　where　k　is　the　maximum　parallelism　degree　and　C　is　the　capacity　of　the　system(i.e.,　the　number　of　machines).　Since　C　＞＞　k　in　BFS,　our　result　significantly　improves　the　known　best　approximation　ratio　of　(C-k/2C-k)(1　-　is　an　element　of)　for　tight　deadlines　[17],　and　C-k/C　.　s-1/s　for　loose　deadlines　[18]　on　a　slackness　ratio　s　＞=　1　that　is　the　maximum　ratio　between　a　job＇s　earliest　actual　finish　time　and　its　deadline.　We　first　propose　feasibility　condition　to　determine　whether　an　instance　of　BFS　is　feasible,　i.e.,　whether　there　exists　a　scheduling　according　to　which　all　jobs　can　finish　before　their　deadlines,　which　is　the　key　to　achieve　the　ratio　improvement　of　our　algorithm.　To　prove　the　correctness　of　the　feasibility　condition,　we　give　a　simple　linear　program(LP)　for　a　weaker　version　of　BFS,　and　show　that　it　is　with　an　integral　polyhedron　and　hence　the　version　of　BFS　is　polynomial-time　solvable.　Then　we　present　a　greedy　algorithm　and　its　equivalent　primal-dual　algorithm　for　the　complementary　problem　of　BFS.　Both　algorithms　have　an　approximation　ratio　of　C-k/C,　and　time　complexity　O(n(2)　+　nT),　where　n　is　the　number　of　jobs　and　T　is　the　number　of　time　slots.　As　a　by-product,　we　show　that　the　BFS　admits　a　polynomial-time　approximation　scheme　(PTAS)　when　T　is　fixed.