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Improved Streaming Algorithms for Maximizing Monotone Submodular Functions under a Knapsack Constraint

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Abstract

In this paper, we consider the problem of maximizing a monotone submodular function subject to a knapsack constraint in a streaming setting. In such a setting, elements arrive sequentially and at any point in time, and the algorithm can store only a small fraction of the elements that have arrived so far. For the special case that all elements have unit sizes (i.e., the cardinality-constraint case), one can find a \((0.5-\varepsilon )\)-approximate solution in \(O(K\varepsilon ^{-1})\) space, where K is the knapsack capacity (Badanidiyuru et al. KDD 2014). The approximation ratio is recently shown to be optimal (Feldman et al. STOC 2020). In this work, we propose a \((0.4-\varepsilon )\)-approximation algorithm for the knapsack-constrained problem, using space that is a polynomial of K and \(\varepsilon \). This improves on the previous best ratio of \(0.363-\varepsilon \) with space of the same order. Our algorithm is based on a careful combination of various ideas to transform multiple-pass streaming algorithms into a single-pass one.

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Notes

  1. The approximation ratio is recently improved to 0.405 [40].

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Acknowledgements

The authors thank the referees for their valuable comments on this manuscript.

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  1. CNRS, École Normale Supérieure, Paris, France

    Chien-Chung Huang

  2. Keio University, Yokohama, Japan

    Naonori Kakimura

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  1. Chien-Chung Huang
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Correspondence to Naonori Kakimura.

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A preliminary version appears in The Algorithms and Data Structures Symposium (WADS) 2019. The first author is supported by ANR-19-CE48-0016 and ANR-18-CE40-0025-01 from the French National Research Agency (ANR). The second author is supported by JSPS KAKENHI Grant Numbers JP17K00028 and JP18H05291.

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Huang, CC., Kakimura, N. Improved Streaming Algorithms for Maximizing Monotone Submodular Functions under a Knapsack Constraint. Algorithmica 83, 879–902 (2021). https://doi.org/10.1007/s00453-020-00786-4

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