Many　combinatorial　optimization　problems　can　be　reduced　to　submodular　optimization　problems.　However,　many　cases　in　practical　applications　do　not　fully　comply　with　the　diminishing　returns　characteristic.　This　paper　studies　the　problem　of　maximizing　weakly-monotone　non-submodular　non-negative　set　functions　without　constraints,　and　extends　the　normal　submodular　ratio　to　weakly-monotone　submodular　ratio　γ^.　In　this　paper,　two　algorithms　are　given　for　the　above　problems.　The　first　one　is　a　deterministic　greedy　approximation　algorithm,　which　realizes　the　approximation　ratio　of　γ^γ^+2.　When　γ^　=　1　,　the　approximate　ratio　is　1/3,　which　matches　the　ratio　of　the　best　deterministic　algorithm　known　for　the　maximization　of　submodular　function　without　constraints.　The　second　algorithm　is　a　random　greedy　algorithm,　which　improves　the　approximation　ratio　to　γ^γ^+1.　When　γ^　=　1　,　the　approximation　ratio　is　1/2,　the　same　as　the　best　algorithm　for　the　maximization　of　unconstrained　submodular　set　functions.　©　2023,　The　Author(s),　under　exclusive　licence　to　Springer　Science+Business　Media,　LLC,　part　of　Springer　Nature.