A Tight Approximation for Submodular Maximization with Mixed Packing and Covering Constraints
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Motivated by applications in machine learning, such as subset selection and data summarization, we consider the problem of maximizing a monotone submodular function given mixed packing and covering constraints. This problem captures numerous classical combinatorial optimization problems, including maximum coverage and submodular knapsack. We present a tight approximation that for any constant ε >0 achieves a guarantee of 1-1/e - ε while violating the covering constraints only by an ε fraction. This generalizes the result of Kulik et al. (SODA'09) for pure packing constraints. Our algorithm is based on a continuous approach that is applied to a non down-monotone polytope along with a novel preprocessing enumeration technique that allows for high concentration bounds in the presence of not only packing but also covering constraints. Additionally, despite the presence of both packing and covering constraints we can still guarantee only a one-sided violation of the covering constraints. We extend the above main result in several directions. First, we consider the case where in addition to the mixed packing and covering constraints a matroid independence constraint is also present. Second, we consider the case where multiple submodular objectives are present and the goal is to find pareto set optimal solutions. Third, we focus on the dependence of the running time of our main algorithm on 1/ε. Whereas previous work on the special case of only packing constraints, as well as our main algorithm, has a running time of n^poly(1/ε), we show a novel purely combinatorial approach that can be applied to several special cases of the problem and reduce the running time to depend polynomially on 1/ε. We believe this new combinatorial approach might be of independent interest.
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