  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.
