Relevance　and　diversity　are　two　desirable　properties　in　data　retrieval　applications,　an　important　field　in　data　science　and　machine　learning.　In　this　paper,　we　consider　three　maximization　problems　to　balance　these　two　factors.　The　objective　function　in　each　problem　is　the　sum　of　a　monotone　submodular　function　f　and　a　supermodular　function　g,　where　f　and　g　capture　the　relevance　and　diversity　of　any　feasible　solution,　respectively.　In　the　first　problem,　we　consider　a　special　supermodular　diversity　function　g　of　a　sum-sum　format　satisfying　the　relaxed　triangle　inequality,　for　which　we　propose　a　greedy-type　approximation　algorithm　with　an　(1　-　1　/　e,　1　/　(2　α))　-bifactor　approximation　ratio,　improving　the　previous　(1　/　(2　α)　,　1　/　(2　α))　-bifactor　approximation　ratio.　In　the　second　problem,　we　consider　an　arbitrary　supermodular　diversity　function　g,　for　which　we　propose　a　distorted　greedy　method　to　give　a　min{1-kfe-1,1-kge-(1-kg)}-approximation　algorithm,　improving　the　previous　kf-1(1-e-kf(1-kg))-approximation　ratio,　where　kf　and　kg　are　the　curvatures　of　the　submodular　function　f　and　the　supermodular　funciton　g,　respectively.　In　the　third　problem,　we　generalize　the　uniform　matroid　constraint　to　the　p　matroid　constraints,　for　which　we　present　a　local　search　algorithm　to　improve　the　previous　1-kg(1-kg)kf+p-approximation　ratio　to　min{p+1-kfp(p+1),(1-kgp+kg(1-kg)2p+(1-kg)2)}.　©　2021,　The　Author(s),　under　exclusive　licence　to　Springer　Science+Business　Media,　LLC,　part　of　Springer　Nature.