Knapsack and Partition are two important additive problems whose fine-grained complexities in the $(1-\varepsilon)$-approximation setting are not yet settled. In this work, we make progress on both problems by giving improved algorithms. - Knapsack can be $(1 - \varepsilon)$-approximated in $\tilde O(n + (1/\varepsilon) ^ {2.2} )$ time, improving the previous $\tilde O(n + (1/\varepsilon) ^ {2.25} )$ by Jin (ICALP'19). There is a known conditional lower bound of $(n+\varepsilon)^{2-o(1)}$ based on $(\min,+)$-convolution hypothesis. - Partition can be $(1 - \varepsilon)$-approximated in $\tilde O(n + (1/\varepsilon) ^ {1.25} )$ time, improving the previous $\tilde O(n + (1/\varepsilon) ^ {1.5} )$ by Bringmann and Nakos (SODA'21). There is a known conditional lower bound of $(1/\varepsilon)^{1-o(1)}$ based on Strong Exponential Time Hypothesis. Both of our new algorithms apply the additive combinatorial results on dense subset sums by Galil and Margalit (SICOMP'91), Bringmann and Wellnitz (SODA'21). Such techniques have not been explored in the context of Knapsack prior to our work. In addition, we design several new methods to speed up the divide-and-conquer steps which naturally arise in solving additive problems.

翻译：Knapscack 和 Parttion 是两个重要的添加问题, 其精密复杂性尚未解决 $( 1-\ varepsilon) $( 1-\ varepsilon) 的精确复杂性尚未解决 。 在这项工作中, 我们通过提供改进的算法, 在这两个问题上都取得进展 。 Knapsack 可以是$( 1 -\ varepsilon) $( $) ( + ( 1 /\ varepsilon) 和 $( 1 + ( \ varepsilon) 美元), 改进以前的 美元( + ( 1 + ( 1/ varepsil) + ( 1/\ vareplilon) ) 美元 (% 2.25} ) 。 根据$( \ valPralPl Pal Palíl ) 的计算方法, 改进了我们之前的 Ralizn 。