Utilizing　approximation　algorithm,　there　has　been　a　large　quantity　of　work　on　optimization　for　submodular　functions　since　the　1970s.　As　a　variant,　k-submodular　function　appears　in　many　fields　to　match　with　the　developing　background,　in　which　the　outputted　sets　changes　from　one　to　k.　Because　of　the　application　of　submodularity,　some　concepts　and　parameters　describing　the　closeness　to　submodular　function　are　generated,　for　example　approximately　submodular　set　function　and　diminishing-return　(DR)　ratio.　In　our　discussion,　the　k-dimension　set　function　with　matroid　constraint　we　will　maximize　may　not　have　access　to　the　submodularity.　However　it　is　an　approximately　non-k-submodular　set　function　which　contains　DR　ratio.　By　the　greedy　technique,　we　obtain　the　approximation　algorithms.　When　the　value　of　the　DR　ratio　is　set　one,　some　known　results　are　covered.