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

Authors Eyal Mizrachi, Roy Schwartz, Joachim Spoerhase , Sumedha Uniyal

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Eyal Mizrachi
  • Computer Science Department, Technion, Haifa 32000, Israel
Roy Schwartz
  • Computer Science Department, Technion, Haifa 32000, Israel
Joachim Spoerhase
  • Department of Computer Science, Aalto University, Espoo, Finland
Sumedha Uniyal
  • Department of Computer Science, Aalto University, Espoo, Finland


Joachim Spoerhase and Sumedha Uniyal thank an anonymous reviewer for pointing them to the fact that Theorem 6 also applies to polytopes that are not down-closed, which makes it possible to apply a randomized rounding approach.

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Eyal Mizrachi, Roy Schwartz, Joachim Spoerhase, and Sumedha Uniyal. A Tight Approximation for Submodular Maximization with Mixed Packing and Covering Constraints. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 85:1-85:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


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-(1/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 previously known enumeration techniques, can handle both packing and covering constraints. We extend the above main result by additionally handling a matroid independence constraint 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. While this approach does not give tight bounds it yields deterministic and in some special cases also considerably faster algorithms. For example, for the well-studied special case of only packing constraints (Kulik et al. [Math. Oper. Res. `13] and Chekuri 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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Submodular optimization and polymatroids
  • Theory of computation → Approximation algorithms analysis
  • submodular function
  • approximation algorithm
  • covering
  • packing


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