The　paper　proposes　the　optimization　problem　of　maximizing　the　sum　of　suBmodular　and　suPermodular　(BP)　functions　with　partial　monotonicity　under　a　streaming　fashion.　In　this　model,　elements　are　randomly　released　from　the　stream　and　the　utility　is　encoded　by　the　sum　of　partial　monotone　suBmodular　and　suPermodular　functions.　The　goal　is　to　determine　whether　a　subset　from　the　stream　of　size　bounded　by　parameter　k　subject　to　the　summarized　utility　is　as　large　as　possible.　In　this　work,　a　threshold-based　streaming　algorithm　is　presented　for　the　BP　maximization　that　attains　a　((1-κ)/(2-κ)-O)-approximation　with　O(1/4\log3(1/\log((2-κ)k/(1-κ)2})　memory　complexity,　where　κ　denotes　the　parameter　of　supermodularity　ratio.　We　further　consider　a　more　general　model　with　fair　constraints　and　present　a　greedy-based　algorithm　that　obtains　the　same　approximation.　We　finally　investigate　this　fair　model　under　the　streaming　fashion　and　provide　a　((1-κ)4/(2-2κ+κ2)2-O)-approximation　algorithm.　　©　1996-2012　Tsinghua　University　Press.