Before revision:  Our contribution is a theoretically guaranteed and widely applicable smoothing framework based on randomization. We prove that our smoothed greedy algorithm almost recovers original approximation guarantees in expectation for the cases of cardinality and \kappa-extensible system constrains.
After revision:  Our contribution is a theoretically guaranteed versatile framework that makes the greedy algorithm for monotone submodular function maximization differentiable. We smooth the greedy algorithm via randomization, and prove that it almost recovers original approximation guarantees in expectation for the cases of cardinality and \kappa-extensible system constrains.