Weakly Approximating Knapsack in Subquadratic Time
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866916701276209152 |
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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 consider the classic Knapsack problem. Let $t$ and $\mathrm{OPT}$ be the capacity and the optimal value, respectively. If one seeks a solution with total profit at least $\mathrm{OPT}/(1 + \varepsilon)$ and total weight at most $t$, then Knapsack can be solved in $\tilde{O}(n + (\frac{1}{\varepsilon})^2)$ time [Chen, Lian, Mao, and Zhang '24][Mao '24]. This running time is the best possible (up to a logarithmic factor), assuming that $(\min,+)$-convolution cannot be solved in truly subquadratic time [Künnemann, Paturi, and Schneider '17][Cygan, Mucha, Węgrzycki, and Włodarczyk '19]. The same upper and lower bounds hold if one seeks a solution with total profit at least $\mathrm{OPT}$ and total weight at most $(1 + \varepsilon)t$. Therefore, it is natural to ask the following question.
If one seeks a solution with total profit at least $\mathrm{OPT}/(1+\varepsilon)$ and total weight at most $(1 + \varepsilon)t$, can Knsapck be solved in $\tilde{O}(n + (\frac{1}{\varepsilon})^{2-δ})$ time for some constant $δ> 0$?
We answer this open question affirmatively by proposing an $\tilde{O}(n + (\frac{1}{\varepsilon})^{7/4})$-time algorithm. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_15001 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Weakly Approximating Knapsack in Subquadratic Time Chen, Lin Lian, Jiayi Mao, Yuchen Zhang, Guochuan Data Structures and Algorithms We consider the classic Knapsack problem. Let $t$ and $\mathrm{OPT}$ be the capacity and the optimal value, respectively. If one seeks a solution with total profit at least $\mathrm{OPT}/(1 + \varepsilon)$ and total weight at most $t$, then Knapsack can be solved in $\tilde{O}(n + (\frac{1}{\varepsilon})^2)$ time [Chen, Lian, Mao, and Zhang '24][Mao '24]. This running time is the best possible (up to a logarithmic factor), assuming that $(\min,+)$-convolution cannot be solved in truly subquadratic time [Künnemann, Paturi, and Schneider '17][Cygan, Mucha, Węgrzycki, and Włodarczyk '19]. The same upper and lower bounds hold if one seeks a solution with total profit at least $\mathrm{OPT}$ and total weight at most $(1 + \varepsilon)t$. Therefore, it is natural to ask the following question. If one seeks a solution with total profit at least $\mathrm{OPT}/(1+\varepsilon)$ and total weight at most $(1 + \varepsilon)t$, can Knsapck be solved in $\tilde{O}(n + (\frac{1}{\varepsilon})^{2-δ})$ time for some constant $δ> 0$? We answer this open question affirmatively by proposing an $\tilde{O}(n + (\frac{1}{\varepsilon})^{7/4})$-time algorithm. |
| title | Weakly Approximating Knapsack in Subquadratic Time |
| topic | Data Structures and Algorithms |
| url | https://arxiv.org/abs/2504.15001 |