Motivated by applications in machine learning, such as subset selection and data summarization, we consider the problem of maximizing a monotone submodular function subject to mixed packing and covering constraints. We present a tight approximation algorithm that for any constant $epsilon >0$ achieves a guarantee of $1-frac{1}{mathrm{e}}-epsilon$ while violating only the covering constraints by a multiplicative factor of $1-epsilon$. Our algorithm is based on a novel enumeration method, which unlike previous known enumeration techniques, can handle both packing and covering constraints. We extend the above main result by additionally handling a matroid independence constraints as well as finding (approximate) pareto set optimal solutions when multiple submodular objectives are present. Finally, we propose a novel and purely combinatorial dynamic programming approach that can be applied to several special cases of the problem yielding not only {em deterministic} but also considerably faster algorithms. For example, for the well studied special case of only packing constraints (Kulik {em et. al.} [Math. Oper. Res. `13] and Chekuri {em et. al.} [FOCS `10]), we are able to present the first deterministic non-trivial approximation algorithm. We believe our new combinatorial approach might be of independent interest.
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