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

We investigate pseudopolynomial-time algorithms for Bounded Knapsack and Bounded Subset Sum. Recent years have seen a growing interest in settling their fine-grained complexity with respect to various parameters. For Bounded Knapsack, the number of items $n$ and the maximum item weight $w_{\max}$ are two of the most natural parameters that have been studied extensively in the literature. 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 conditional lower bound of $O((n + w_{\max})^{2-o(1)})$ based on $(\min,+)$-convolution hypothesis [Cygan, Mucha, Wegrzycki, Wlodarczyk '17]. We narrow the gap significantly 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. Prior to our work, the best running times (in terms of $n$ and $w_{\max}$) for Bounded 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 by Bringmann '19 and Bringmann, Wellnitz '21], where $\mu_{\max}$ refers to the maximum multiplicity of item weights.