We　investigate　a　non-submodular　maximization　problem　subject　to　a　p-independence　system　constraint,　where　the　non-submodularity　of　the　utility　function　is　characterized　by　a　series　of　parameters,　such　as　submodularity　(supmodularity)　ratio,　generalized　curvature,　and　zero　order　approximate　submodularity　coefficient,　etc.　Inspired　by　Feldman　et　al.　[15]　who　consider　a　non-monotone　submodular　maximization　with　a　p-independence　system　constraint,　we　extend　their　Repeat-Greedy　algorithm　to　non-submodular　setting.　While　there　is　no　general　reduction　to　convert　algorithms　for　submodular　optimization　problems　to　non-submodular　optimization　problems,　we　are　able　to　show　the　extended　Repeat-Greedy　algorithm　has　an　almost　constant　approximation　ratio　for　non-monotone　non-submodular　maximization.　©　Springer　Nature　Switzerland　AG　2019.