Before revision:  We further obtain a tight (1-e^{-1-\epsilon)-approximation for monotone submodular optimization subject to a constant number of multiple knapsack constraints and a matroid constraint, and a (}  0.385- \epsilon )-approximation for non-monotone submodular optimization subject to a constant number of multiple knapsack constraints. At the heart of our algorithms lies a novel representation of a multiple knapsack constraint as a polytope. We consider this key tool as a main technical contribution of this paper .
After revision:  We further obtain a tight (1-e^{-1-\epsilon)-approximation for monotone submodular optimization subject to a constant number of multiple knapsack constraints and a matroid constraint, and a (} 0.385- \epsilon )-approximation , matching the best known ratio for a single knapsack constraint. The robustness of our algorithm is achieved by applying a novel fractional variant of the classical linear grouping technique, which is of independent interest .