Arising　from　practical　problems　such　as　in　sensor　placement　and　influence　maximization　in　social　network,　submodular　and　non-submodular　maximization　on　the　integer　lattice　has　attracted　much　attention　recently.　In　this　work,　we　consider　the　problem　of　maximizing　the　sum　of　a　monotone　non-negative　diminishing　return　submodular　(DR-submodular)　function　and　a　supermodular　function　on　the　integer　lattice　subject　to　a　cardinality　constraint.　By　exploiting　the　special　combinatorial　structures　in　the　problem,　we　introduce　a　decreasing　threshold　greedy　algorithm　with　a　binary　search　as　its　subroutine　to　solve　the　problem.　To　avoid　introducing　the　diminishing　return　ratio　and　submodularity　ratio　of　the　objective　function,　we　generalize　the　total　curvatures　of　submodular　functions　and　supermodular　functions　to　the　integer　lattice　version.　We　show　that　our　algorithm　has　a　constant　approximation　ratio　parameterized　by　the　new　introduced　total　curvatures　on　integer　lattice　with　a　polynomial　query　complexity.