In　this　paper,　we　consider　the　problem　of　maximizing　the　sum　of　a　submodular　and　a　supermodular　(BP)　function　(both　are　non-negative)　under　cardinality　constraint　and　p-system　constraint　respectively,　which　arises　in　many　real-world　applications　such　as　data　science,　machine　learning　and　artificial　intelligence.　Greedy　algorithm　is　widely　used　to　design　an　approximation　algorithm.　However,　in　many　applications,　evaluating　the　value　of　the　objective　function　is　expensive.　In　order　to　avoid　a　waste　of　time　and　money,　we　propose　a　Stochastic-Greedy　(SG)　algorithm,　a　Stochastic-Standard-Greedy　(SSG)　algorithm　as　well　as　a　Random-Greedy　(RG)　for　the　monotone　BP　maximization　problems　under　cardinality　constraint,　p-system　constraint　as　well　as　the　non-monotone　BP　maximization　problems　under　cardinality　constraint,　respectively.　The　SSG　algorithm　also　works　well　on　the　monotone　BP　maximization　problems　under　cardinality　constraint.　Numerical　experiments　for　the　monotone　BP　maximization　under　cardinality　constraint　is　made　for　comparing　the　SG　algorithm　and　the　SSG　algorithm　in　the　previous　works.　The　results　show　that　the　guarantee　of　the　SG　algorithm　is　worse　than　the　SSG　algorithm,　but　the　SG　algorithm　is　faster　than　SSG　algorithm,　especially　for　the　large-scale　instances.　©　2020,　Springer　Nature　Switzerland　AG.