A Tight Approximation for Submodular Maximization with Mixed Packing and Covering Constraints

Eyal Mizrachi; Roy Schwartz; Joachim Spoerhase; Sumedha Uniyal

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arXiv · 1804.10947

A Tight Approximation for Submodular Maximization with Mixed Packing and Covering Constraints

Eyal Mizrachi·Roy Schwartz·Joachim Spoerhase·Sumedha Uniyal

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Abstract

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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Eyal Mizrachi, Roy Schwartz, Joachim Spoerhase, Sumedha Uniyal. 2018-04-29. A Tight Approximation for Submodular Maximization with Mixed Packing and Covering Constraints. https://arxiv.org/abs/1804.10947

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