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arXiv · 2308.07821

A Nearly Quadratic-Time FPTAS for Knapsack

Lin Chen·Jiayi Lian·Yuchen Mao·Guochuan Zhang
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Abstract

We investigate the classic Knapsack problem and propose a fully polynomial-time approximation scheme (FPTAS) that runs in $\widetilde{O}(n + (1/\varepsilon)^2)$ time. This improves upon the $\widetilde{O}(n + (1/\varepsilon)^{11/5})$-time algorithm by Deng, Jin, and Mao [\textit{Proceedings of the 2023 Annual ACM-SIAM Symposium on Discrete Algorithms, 2023}]. Our algorithm is the best possible (up to a polylogarithmic factor) conditioned on the conjecture that $(\min, +)$-convolution has no truly subquadratic-time algorithm, since this conjecture implies that Knapsack has no $O((n + 1/\varepsilon)^{2-\delta})$-time FPTAS for any constant $\delta > 0$.

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Lin Chen, Jiayi Lian, Yuchen Mao, Guochuan Zhang. 2023-08-15. A Nearly Quadratic-Time FPTAS for Knapsack. https://arxiv.org/abs/2308.07821

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