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We investigate pseudopolynomial-time algorithms for Bounded Knapsack andBounded Subset Sum. Recent years have seen a growing interest in settling theirfine-grained complexity with respect to various parameters. For BoundedKnapsack, the number of items $n$ and the maximum item weight $w_{\max}$ aretwo of the most natural parameters that have been studied extensively in theliterature. The previous best running time in terms of $n$ and $w_{\max}$ is$O(n + w^3_{\max})$ [Polak, Rohwedder, Wegrzycki '21]. There is a conditionallower bound of $O((n + w_{\max})^{2-o(1)})$ based on $(\min,+)$-convolutionhypothesis [Cygan, Mucha, Wegrzycki, Wlodarczyk '17]. We narrow the gapsignificantly by proposing a $\tilde{O}(n + w^{12/5}_{\max})$-time algorithm.Note that in the regime where $w_{\max} \approx n$, our algorithm runs in$\tilde{O}(n^{12/5})$ time, while all the previous algorithms require$\Omega(n^3)$ time in the worst case.  For Bounded Subset Sum, we give two algorithms running in$\tilde{O}(nw_{\max})$ and $\tilde{O}(n + w^{3/2}_{\max})$ time, respectively.These results match the currently best running time for 0-1 Subset Sum. Priorto our work, the best running times (in terms of $n$ and $w_{\max}$) forBounded Subset Sum is $\tilde{O}(n + w^{5/3}_{\max})$ [Polak, Rohwedder,Wegrzycki '21] and $\tilde{O}(n + \mu_{\max}^{1/2}w_{\max}^{3/2})$ [implied byBringmann '19 and Bringmann, Wellnitz '21], where $\mu_{\max}$ refers to themaximum multiplicity of item weights.

Faster Algorithms for Bounded Knapsack and Bounded Subset Sum Via Fine-Grained Proximity Results