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Even Faster Knapsack via Rectangular Monotone Min-Plus Convolution and Balancing

Authors Karl Bringmann , Anita Dürr , Adam Polak

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LIPIcs.ESA.2024.33.pdf
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ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
Keywords
  • 0-1-Knapsack problem
  • bounded monotone min-plus convolution
  • fine-grained complexity

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Abstract

We present a pseudopolynomial-time algorithm for the Knapsack problem that has running time Õ(n + t√{p_{max}}), where n is the number of items, t is the knapsack capacity, and p_{max} is the maximum item profit. This improves over the Õ(n + t p_{max})-time algorithm based on the convolution and prediction technique by Bateni et al. (STOC 2018). Moreover, we give some evidence, based on a strengthening of the Min-Plus Convolution Hypothesis, that our running time might be optimal.
Our algorithm uses two new technical tools, which might be of independent interest. First, we generalize the Õ(n^{1.5})-time algorithm for bounded monotone min-plus convolution by Chi et al. (STOC 2022) to the rectangular case where the range of entries can be different from the sequence length. Second, we give a reduction from general knapsack instances to balanced instances, where all items have nearly the same profit-to-weight ratio, up to a constant factor.
Using these techniques, we can also obtain algorithms that run in time Õ(n + OPT√{w_{max}}), Õ(n + (nw_{max}p_{max})^{1/3}t^{2/3}), and Õ(n + (nw_{max}p_{max})^{1/3} OPT^{2/3}), where OPT is the optimal total profit and w_{max} is the maximum item weight.

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Karl Bringmann, Anita Dürr, and Adam Polak. Even Faster Knapsack via Rectangular Monotone Min-Plus Convolution and Balancing. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 33:1-33:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.ESA.2024.33

Author Details

Karl Bringmann
  • Saarland University and Max Planck Institute for Informatics, Saarland Informatics Campus, Saarbrücken, Germany
Anita Dürr
  • Saarland University and Max Planck Institute for Informatics, Saarland Informatics Campus, Saarbrücken, Germany
Adam Polak
  • Bocconi University, Milan, Italy

Funding

Karl Bringmann and Anita Dürr: This work is part of the project TIPEA that has received funding from the European Research Council (ERC) under the European Unions Horizon 2020 research and innovation programme (grant agreement No. 850979).
  • Polak, Adam: Part of this work was done when Adam Polak was at Max Planck Institute for Informatics.

Acknowledgements

The authors thank Alejandro Cassis for many fruitful discussions.

References

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