A Nearly Quadratic-Time FPTAS for Knapsack
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866913638340624384 |
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| author | Chen, Lin Lian, Jiayi Mao, Yuchen Zhang, Guochuan |
| author_facet | Chen, Lin Lian, Jiayi Mao, Yuchen Zhang, Guochuan |
| contents | 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 $O((n + 1/\varepsilon)^{2-δ})$-time FPTAS for any constant $δ> 0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_07821 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A Nearly Quadratic-Time FPTAS for Knapsack Chen, Lin Lian, Jiayi Mao, Yuchen Zhang, Guochuan Data Structures and Algorithms 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 $O((n + 1/\varepsilon)^{2-δ})$-time FPTAS for any constant $δ> 0$. |
| title | A Nearly Quadratic-Time FPTAS for Knapsack |
| topic | Data Structures and Algorithms |
| url | https://arxiv.org/abs/2308.07821 |