0-1 Knapsack in Nearly Quadratic Time
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Author(s)
Jin, Ce
Date Issued
June 10, 2024
Publisher
Association for Computing Machinery STOC 2024: Proceedings of the 56th Annual ACM Symposium on Theory of Computing
Citation
Jin, Ce. 2024. "0-1 Knapsack in Nearly Quadratic Time."
Version
Final published version
Abstract
We study pseudo-polynomial time algorithms for the fundamental 0-
1 Knapsack problem. Recent research interest has focused on its finegrained complexity with respect to the number of items 𝑛 and the
maximum item weight 𝑤max. Under (min, +)-convolution hypothesis, 0-1 Knapsack does not have𝑂( (𝑛+𝑤max)
2−𝛿
) time algorithms
(Cygan-Mucha-Węgrzycki-Włodarczyk 2017 and Künnemann-PaturiSchneider 2017). On the upper bound side, currently the fastest
algorithm runs in 𝑂e(𝑛 + 𝑤
12/5
max ) time (Chen, Lian, Mao, and Zhang
2023), improving the earlier 𝑂(𝑛 + 𝑤
3
max)-time algorithm by Polak,
Rohwedder, and Węgrzycki (2021).
In this paper, we close this gap between the upper bound and
the conditional lower bound (up to subpolynomial factors): The
0-1 Knapsack problem has a deterministic algorithm in 𝑂(𝑛 +
𝑤
2
max log4 𝑤max) time.
Our algorithm combines and extends several recent structural
results and algorithmic techniques from the literature on knapsacktype problems:
(1) We generalize the “fine-grained proximity” technique of
Chen, Lian, Mao, and Zhang (2023) derived from the additivecombinatorial results of Bringmann and Wellnitz (2021) on dense
subset sums. This allows us to bound the support size of the useful
partial solutions in the dynamic program.
(2) To exploit the small support size, our main technical component is a vast extension of the “witness propagation” method,
originally designed by Deng, Mao, and Zhong (2023) for speeding up
dynamic programming in the easier unbounded knapsack settings.
To extend this approach to our 0-1 setting, we use a novel pruning
method, as well as the two-level color-coding of Bringmann (2017)
and the SMAWK algorithm on tall matrices.
MIT Department
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
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DOI of Published Version
https://doi.org/10.1145/3618260.3649618
